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Theorem intirr 4654
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 3123 . . . 4 (𝑅 ∩ I ) = ( I ∩ 𝑅)
21eqeq1i 2044 . . 3 ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅)
3 disj2 3269 . . 3 (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅))
4 reli 4408 . . . 4 Rel I
5 ssrel 4371 . . . 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 1590 . . . . 5 (y = xx = y)
9 vex 2554 . . . . . 6 y V
109ideq 4431 . . . . 5 (x I yx = y)
11 df-br 3756 . . . . 5 (x I y ↔ ⟨x, y I )
128, 10, 113bitr2i 197 . . . 4 (y = x ↔ ⟨x, y I )
13 vex 2554 . . . . . . . 8 x V
1413, 9opex 3957 . . . . . . 7 x, y V
1514biantrur 287 . . . . . 6 (¬ ⟨x, y 𝑅 ↔ (⟨x, y V ¬ ⟨x, y 𝑅))
16 eldif 2921 . . . . . 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 3756 . . . . 5 (x𝑅y ↔ ⟨x, y 𝑅)
1917, 18xchnxbir 605 . . . 4 x𝑅y ↔ ⟨x, y (V ∖ 𝑅))
2012, 19imbi12i 228 . . 3 ((y = x → ¬ x𝑅y) ↔ (⟨x, y I → ⟨x, y (V ∖ 𝑅)))
21202albii 1357 . 2 (xy(y = x → ¬ x𝑅y) ↔ xy(⟨x, y I → ⟨x, y (V ∖ 𝑅)))
22 nfv 1418 . . . 4 y ¬ x𝑅x
23 breq2 3759 . . . . 5 (y = x → (x𝑅yx𝑅x))
2423notbid 591 . . . 4 (y = x → (¬ x𝑅y ↔ ¬ x𝑅x))
2522, 24equsal 1612 . . 3 (y(y = x → ¬ x𝑅y) ↔ ¬ x𝑅x)
2625albii 1356 . 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 1240   = wceq 1242   wcel 1390  Vcvv 2551  cdif 2908  cin 2910  wss 2911  c0 3218  cop 3370   class class class wbr 3755   I cid 4016  Rel wrel 4293
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295
This theorem is referenced by: (None)
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