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Theorem intirr 6140
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 4216 . . . 4 (𝑅 ∩ I ) = ( I ∩ 𝑅)
21eqeq1i 2739 . . 3 ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅)
3 disj2 4463 . . 3 (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅))
4 reli 5838 . . . 4 Rel I
5 ssrel 5794 . . . 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 297 . 2 ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
8 equcom 2014 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
9 vex 3481 . . . . . 6 𝑦 ∈ V
109ideq 5865 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
11 df-br 5148 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
128, 10, 113bitr2i 299 . . . 4 (𝑦 = 𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
13 opex 5474 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
1413biantrur 530 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
15 eldif 3972 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1614, 15bitr4i 278 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
17 df-br 5148 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1816, 17xchnxbir 333 . . . 4 𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
1912, 18imbi12i 350 . . 3 ((𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
20192albii 1816 . 2 (∀𝑥𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
21 breq2 5151 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
2221notbid 318 . . . 4 (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
2322equsalvw 2000 . . 3 (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥)
2423albii 1815 . 2 (∀𝑥𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)
257, 20, 243bitr2i 299 1 ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1534   = wceq 1536  wcel 2105  Vcvv 3477  cdif 3959  cin 3961  wss 3962  c0 4338  cop 4636   class class class wbr 5147   I cid 5581  Rel wrel 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695
This theorem is referenced by:  hartogslem1  9579  hausdiag  23668
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