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Theorem intirr 6125
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 4199 . . . 4 (𝑅 ∩ I ) = ( I ∩ 𝑅)
21eqeq1i 2730 . . 3 ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅)
3 disj2 4459 . . 3 (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅))
4 reli 5828 . . . 4 Rel I
5 ssrel 5784 . . . 4 (Rel I → ( I ⊆ (V ∖ 𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))))
64, 5ax-mp 5 . . 3 ( I ⊆ (V ∖ 𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
72, 3, 63bitri 296 . 2 ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
8 equcom 2013 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
9 vex 3465 . . . . . 6 𝑦 ∈ V
109ideq 5855 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
11 df-br 5150 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
128, 10, 113bitr2i 298 . . . 4 (𝑦 = 𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
13 opex 5466 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
1413biantrur 529 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
15 eldif 3954 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1614, 15bitr4i 277 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
17 df-br 5150 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1816, 17xchnxbir 332 . . . 4 𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
1912, 18imbi12i 349 . . 3 ((𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
20192albii 1814 . 2 (∀𝑥𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
21 breq2 5153 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
2221notbid 317 . . . 4 (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
2322equsalvw 1999 . . 3 (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥)
2423albii 1813 . 2 (∀𝑥𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)
257, 20, 243bitr2i 298 1 ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wcel 2098  Vcvv 3461  cdif 3941  cin 3943  wss 3944  c0 4322  cop 4636   class class class wbr 5149   I cid 5575  Rel wrel 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685
This theorem is referenced by:  hartogslem1  9567  hausdiag  23593
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