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Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddge0 7201 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((A B ℝ) (0 ≤ A 0 ≤ B)) → 0 ≤ (A + B))
 
Theoremltaddpos 7202 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
((A B ℝ) → (0 < AB < (B + A)))
 
Theoremltaddpos2 7203 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
((A B ℝ) → (0 < AB < (A + B)))
 
Theoremltsubpos 7204 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((A B ℝ) → (0 < A ↔ (BA) < B))
 
Theoremposdif 7205 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
((A B ℝ) → (A < B ↔ 0 < (BA)))
 
Theoremlesub1 7206 Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (AB ↔ (A𝐶) ≤ (B𝐶)))
 
Theoremlesub2 7207 Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (AB ↔ (𝐶B) ≤ (𝐶A)))
 
Theoremltsub1 7208 Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (A < B ↔ (A𝐶) < (B𝐶)))
 
Theoremltsub2 7209 Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (A < B ↔ (𝐶B) < (𝐶A)))
 
Theoremlt2sub 7210 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A < 𝐶 𝐷 < B) → (AB) < (𝐶𝐷)))
 
Theoremle2sub 7211 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A𝐶 𝐷B) → (AB) ≤ (𝐶𝐷)))
 
Theoremltneg 7212 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 27-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℝ) → (A < B ↔ -B < -A))
 
Theoremltnegcon1 7213 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
((A B ℝ) → (-A < B ↔ -B < A))
 
Theoremltnegcon2 7214 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
((A B ℝ) → (A < -BB < -A))
 
Theoremleneg 7215 Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℝ) → (AB ↔ -B ≤ -A))
 
Theoremlenegcon1 7216 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
((A B ℝ) → (-AB ↔ -BA))
 
Theoremlenegcon2 7217 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
((A B ℝ) → (A ≤ -BB ≤ -A))
 
Theoremlt0neg1 7218 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
(A ℝ → (A < 0 ↔ 0 < -A))
 
Theoremlt0neg2 7219 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
(A ℝ → (0 < A ↔ -A < 0))
 
Theoremle0neg1 7220 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
(A ℝ → (A ≤ 0 ↔ 0 ≤ -A))
 
Theoremle0neg2 7221 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
(A ℝ → (0 ≤ A ↔ -A ≤ 0))
 
Theoremaddge01 7222 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
((A B ℝ) → (0 ≤ BA ≤ (A + B)))
 
Theoremaddge02 7223 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
((A B ℝ) → (0 ≤ BA ≤ (B + A)))
 
Theoremadd20 7224 Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((A 0 ≤ A) (B 0 ≤ B)) → ((A + B) = 0 ↔ (A = 0 B = 0)))
 
Theoremsubge0 7225 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℝ) → (0 ≤ (AB) ↔ BA))
 
Theoremsuble0 7226 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℝ) → ((AB) ≤ 0 ↔ AB))
 
Theoremleaddle0 7227 The sum of a real number and a second real number is less then the real number iff the second real number is negative. (Contributed by Alexander van der Vekens, 30-May-2018.)
((A B ℝ) → ((A + B) ≤ AB ≤ 0))
 
Theoremsubge02 7228 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
((A B ℝ) → (0 ≤ B ↔ (AB) ≤ A))
 
Theoremlesub0 7229 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℝ) → ((0 ≤ A B ≤ (BA)) ↔ A = 0))
 
Theoremmullt0 7230 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
(((A A < 0) (B B < 0)) → 0 < (A · B))
 
Theorem0le1 7231 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
0 ≤ 1
 
Theoremleidi 7232 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
A        AA
 
Theoremgt0ne0i 7233 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
A        (0 < AA ≠ 0)
 
Theoremgt0ne0ii 7234 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
A     &   0 < A       A ≠ 0
 
Theoremaddgt0i 7235 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B        ((0 < A 0 < B) → 0 < (A + B))
 
Theoremaddge0i 7236 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B        ((0 ≤ A 0 ≤ B) → 0 ≤ (A + B))
 
Theoremaddgegt0i 7237 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
A     &   B        ((0 ≤ A 0 < B) → 0 < (A + B))
 
Theoremaddgt0ii 7238 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)
A     &   B     &   0 < A    &   0 < B       0 < (A + B)
 
Theoremadd20i 7239 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → ((A + B) = 0 ↔ (A = 0 B = 0)))
 
Theoremltnegi 7240 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
A     &   B        (A < B ↔ -B < -A)
 
Theoremlenegi 7241 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
A     &   B        (AB ↔ -B ≤ -A)
 
Theoremltnegcon2i 7242 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
A     &   B        (A < -BB < -A)
 
Theoremlesub0i 7243 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B        ((0 ≤ A B ≤ (BA)) ↔ A = 0)
 
Theoremltaddposi 7244 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
A     &   B        (0 < AB < (B + A))
 
Theoremposdifi 7245 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
A     &   B        (A < B ↔ 0 < (BA))
 
Theoremltnegcon1i 7246 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
A     &   B        (-A < B ↔ -B < A)
 
Theoremlenegcon1i 7247 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
A     &   B        (-AB ↔ -BA)
 
Theoremsubge0i 7248 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
A     &   B        (0 ≤ (AB) ↔ BA)
 
Theoremltadd1i 7249 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
A     &   B     &   𝐶        (A < B ↔ (A + 𝐶) < (B + 𝐶))
 
Theoremleadd1i 7250 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
A     &   B     &   𝐶        (AB ↔ (A + 𝐶) ≤ (B + 𝐶))
 
Theoremleadd2i 7251 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
A     &   B     &   𝐶        (AB ↔ (𝐶 + A) ≤ (𝐶 + B))
 
Theoremltsubaddi 7252 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B     &   𝐶        ((AB) < 𝐶A < (𝐶 + B))
 
Theoremlesubaddi 7253 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B     &   𝐶        ((AB) ≤ 𝐶A ≤ (𝐶 + B))
 
Theoremltsubadd2i 7254 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
A     &   B     &   𝐶        ((AB) < 𝐶A < (B + 𝐶))
 
Theoremlesubadd2i 7255 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
A     &   B     &   𝐶        ((AB) ≤ 𝐶A ≤ (B + 𝐶))
 
Theoremltaddsubi 7256 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((A + B) < 𝐶A < (𝐶B))
 
Theoremlt2addi 7257 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶     &   𝐷        ((A < 𝐶 B < 𝐷) → (A + B) < (𝐶 + 𝐷))
 
Theoremle2addi 7258 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
A     &   B     &   𝐶     &   𝐷        ((A𝐶 B𝐷) → (A + B) ≤ (𝐶 + 𝐷))
 
Theoremgt0ne0d 7259 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(φ → 0 < A)       (φA ≠ 0)
 
Theoremlt0ne0d 7260 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
(φA < 0)       (φA ≠ 0)
 
Theoremleidd 7261 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φAA)
 
Theoremlt0neg1d 7262 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → (A < 0 ↔ 0 < -A))
 
Theoremlt0neg2d 7263 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → (0 < A ↔ -A < 0))
 
Theoremle0neg1d 7264 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → (A ≤ 0 ↔ 0 ≤ -A))
 
Theoremle0neg2d 7265 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → (0 ≤ A ↔ -A ≤ 0))
 
Theoremaddgegt0d 7266 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 ≤ A)    &   (φ → 0 < B)       (φ → 0 < (A + B))
 
Theoremaddgt0d 7267 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 < A)    &   (φ → 0 < B)       (φ → 0 < (A + B))
 
Theoremaddge0d 7268 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 ≤ A)    &   (φ → 0 ≤ B)       (φ → 0 ≤ (A + B))
 
Theoremltnegd 7269 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A < B ↔ -B < -A))
 
Theoremlenegd 7270 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (AB ↔ -B ≤ -A))
 
Theoremltnegcon1d 7271 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → -A < B)       (φ → -B < A)
 
Theoremltnegcon2d 7272 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φA < -B)       (φB < -A)
 
Theoremlenegcon1d 7273 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → -AB)       (φ → -BA)
 
Theoremlenegcon2d 7274 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φA ≤ -B)       (φB ≤ -A)
 
Theoremltaddposd 7275 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 < AB < (B + A)))
 
Theoremltaddpos2d 7276 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 < AB < (A + B)))
 
Theoremltsubposd 7277 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 < A ↔ (BA) < B))
 
Theoremposdifd 7278 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A < B ↔ 0 < (BA)))
 
Theoremaddge01d 7279 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 ≤ BA ≤ (A + B)))
 
Theoremaddge02d 7280 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 ≤ BA ≤ (B + A)))
 
Theoremsubge0d 7281 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 ≤ (AB) ↔ BA))
 
Theoremsuble0d 7282 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → ((AB) ≤ 0 ↔ AB))
 
Theoremsubge02d 7283 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 ≤ B ↔ (AB) ≤ A))
 
Theoremltadd1d 7284 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (A < B ↔ (A + 𝐶) < (B + 𝐶)))
 
Theoremleadd1d 7285 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (A + 𝐶) ≤ (B + 𝐶)))
 
Theoremleadd2d 7286 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (𝐶 + A) ≤ (𝐶 + B)))
 
Theoremltsubaddd 7287 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) < 𝐶A < (𝐶 + B)))
 
Theoremlesubaddd 7288 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) ≤ 𝐶A ≤ (𝐶 + B)))
 
Theoremltsubadd2d 7289 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) < 𝐶A < (B + 𝐶)))
 
Theoremlesubadd2d 7290 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) ≤ 𝐶A ≤ (B + 𝐶)))
 
Theoremltaddsubd 7291 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((A + B) < 𝐶A < (𝐶B)))
 
Theoremltaddsub2d 7292 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((A + B) < 𝐶B < (𝐶A)))
 
Theoremleaddsub2d 7293 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((A + B) ≤ 𝐶B ≤ (𝐶A)))
 
Theoremsubled 7294 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ → (AB) ≤ 𝐶)       (φ → (A𝐶) ≤ B)
 
Theoremlesubd 7295 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA ≤ (B𝐶))       (φ𝐶 ≤ (BA))
 
Theoremltsub23d 7296 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ → (AB) < 𝐶)       (φ → (A𝐶) < B)
 
Theoremltsub13d 7297 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < (B𝐶))       (φ𝐶 < (BA))
 
Theoremlesub1d 7298 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (A𝐶) ≤ (B𝐶)))
 
Theoremlesub2d 7299 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (𝐶B) ≤ (𝐶A)))
 
Theoremltsub1d 7300 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (A < B ↔ (A𝐶) < (B𝐶)))
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