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Theorem intirr 6138
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 4209 . . . 4 (𝑅 ∩ I ) = ( I ∩ 𝑅)
21eqeq1i 2742 . . 3 ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅)
3 disj2 4458 . . 3 (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅))
4 reli 5836 . . . 4 Rel I
5 ssrel 5792 . . . 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 2017 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
9 vex 3484 . . . . . 6 𝑦 ∈ V
109ideq 5863 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
11 df-br 5144 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
128, 10, 113bitr2i 299 . . . 4 (𝑦 = 𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
13 opex 5469 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
1413biantrur 530 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
15 eldif 3961 . . . . . 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 5144 . . . . 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 1820 . 2 (∀𝑥𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
21 breq2 5147 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
2221notbid 318 . . . 4 (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
2322equsalvw 2003 . . 3 (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥)
2423albii 1819 . 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 1538   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  cin 3950  wss 3951  c0 4333  cop 4632   class class class wbr 5143   I cid 5577  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692
This theorem is referenced by:  hartogslem1  9582  hausdiag  23653
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