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Theorem intirr 4711
 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 3129 . . . 4 (𝑅 ∩ I ) = ( I ∩ 𝑅)
21eqeq1i 2047 . . 3 ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅)
3 disj2 3275 . . 3 (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅))
4 reli 4465 . . . 4 Rel I
5 ssrel 4428 . . . 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 1593 . . . . 5 (𝑦 = 𝑥𝑥 = 𝑦)
9 vex 2560 . . . . . 6 𝑦 ∈ V
109ideq 4488 . . . . 5 (𝑥 I 𝑦𝑥 = 𝑦)
11 df-br 3765 . . . . 5 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
128, 10, 113bitr2i 197 . . . 4 (𝑦 = 𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
13 vex 2560 . . . . . . . 8 𝑥 ∈ V
1413, 9opex 3966 . . . . . . 7 𝑥, 𝑦⟩ ∈ V
1514biantrur 287 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
16 eldif 2927 . . . . . 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 3765 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
1917, 18xchnxbir 606 . . . 4 𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
2012, 19imbi12i 228 . . 3 ((𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
21202albii 1360 . 2 (∀𝑥𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
22 nfv 1421 . . . 4 𝑦 ¬ 𝑥𝑅𝑥
23 breq2 3768 . . . . 5 (𝑦 = 𝑥 → (𝑥𝑅𝑦𝑥𝑅𝑥))
2423notbid 592 . . . 4 (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
2522, 24equsal 1615 . . 3 (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥)
2625albii 1359 . 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 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∖ cdif 2914   ∩ cin 2916   ⊆ wss 2917  ∅c0 3224  ⟨cop 3378   class class class wbr 3764   I cid 4025  Rel wrel 4350 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352 This theorem is referenced by: (None)
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