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Theorem intirr 5976
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 4182 . . . 4 (𝑅 ∩ I ) = ( I ∩ 𝑅)
21eqeq1i 2831 . . 3 ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅)
3 disj2 4410 . . 3 (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅))
4 reli 5697 . . . 4 Rel I
5 ssrel 5656 . . . 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 298 . 2 ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
8 equcom 2018 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
9 vex 3503 . . . . . 6 𝑦 ∈ V
109ideq 5722 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
11 df-br 5064 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
128, 10, 113bitr2i 300 . . . 4 (𝑦 = 𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
13 opex 5353 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
1413biantrur 531 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
15 eldif 3950 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
1614, 15bitr4i 279 . . . . 5 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
17 df-br 5064 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1816, 17xchnxbir 334 . . . 4 𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
1912, 18imbi12i 352 . . 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 5067 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
2221notbid 319 . . . 4 (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
2322equsalvw 2003 . . 3 (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥)
2423albii 1813 . 2 (∀𝑥𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)
257, 20, 243bitr2i 300 1 ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1528   = wceq 1530  wcel 2107  Vcvv 3500  cdif 3937  cin 3939  wss 3940  c0 4295  cop 4570   class class class wbr 5063   I cid 5458  Rel wrel 5559
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561
This theorem is referenced by:  hartogslem1  8995  hausdiag  22169
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