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Theorem intirr 4638
Description: Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
intirr ((𝑅 ∩ I ) = ∅ ↔ x ¬ x𝑅x)
Distinct variable group:   x,𝑅

Proof of Theorem intirr
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 incom 3106 . . . 4 (𝑅 ∩ I ) = ( I ∩ 𝑅)
21eqeq1i 2029 . . 3 ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅)
3 disj2 3252 . . 3 (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅))
4 reli 4392 . . . 4 Rel I
5 ssrel 4355 . . . 4 (Rel I → ( I ⊆ (V ∖ 𝑅) ↔ xy(⟨x, y I → ⟨x, y (V ∖ 𝑅))))
64, 5ax-mp 7 . . 3 ( I ⊆ (V ∖ 𝑅) ↔ xy(⟨x, y I → ⟨x, y (V ∖ 𝑅)))
72, 3, 63bitri 195 . 2 ((𝑅 ∩ I ) = ∅ ↔ xy(⟨x, y I → ⟨x, y (V ∖ 𝑅)))
8 equcom 1575 . . . . 5 (y = xx = y)
9 vex 2538 . . . . . 6 y V
109ideq 4415 . . . . 5 (x I yx = y)
11 df-br 3739 . . . . 5 (x I y ↔ ⟨x, y I )
128, 10, 113bitr2i 197 . . . 4 (y = x ↔ ⟨x, y I )
13 vex 2538 . . . . . . . 8 x V
1413, 9opex 3940 . . . . . . 7 x, y V
1514biantrur 287 . . . . . 6 (¬ ⟨x, y 𝑅 ↔ (⟨x, y V ¬ ⟨x, y 𝑅))
16 eldif 2904 . . . . . 6 (⟨x, y (V ∖ 𝑅) ↔ (⟨x, y V ¬ ⟨x, y 𝑅))
1715, 16bitr4i 176 . . . . 5 (¬ ⟨x, y 𝑅 ↔ ⟨x, y (V ∖ 𝑅))
18 df-br 3739 . . . . 5 (x𝑅y ↔ ⟨x, y 𝑅)
1917, 18xchnxbir 593 . . . 4 x𝑅y ↔ ⟨x, y (V ∖ 𝑅))
2012, 19imbi12i 228 . . 3 ((y = x → ¬ x𝑅y) ↔ (⟨x, y I → ⟨x, y (V ∖ 𝑅)))
21202albii 1340 . 2 (xy(y = x → ¬ x𝑅y) ↔ xy(⟨x, y I → ⟨x, y (V ∖ 𝑅)))
22 nfv 1402 . . . 4 y ¬ x𝑅x
23 breq2 3742 . . . . 5 (y = x → (x𝑅yx𝑅x))
2423notbid 579 . . . 4 (y = x → (¬ x𝑅y ↔ ¬ x𝑅x))
2522, 24equsal 1597 . . 3 (y(y = x → ¬ x𝑅y) ↔ ¬ x𝑅x)
2625albii 1339 . 2 (xy(y = x → ¬ x𝑅y) ↔ x ¬ x𝑅x)
277, 21, 263bitr2i 197 1 ((𝑅 ∩ I ) = ∅ ↔ x ¬ x𝑅x)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98  wal 1226   = wceq 1228   wcel 1374  Vcvv 2535  cdif 2891  cin 2893  wss 2894  c0 3201  cop 3353   class class class wbr 3738   I cid 3999  Rel wrel 4277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279
This theorem is referenced by: (None)
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