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Theorem epinid0 9056
Description: The membership (epsilon) relation and the identity relation are disjoint. Variable-free version of nelaneq 9055. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 18-Jun-2022.)
Assertion
Ref Expression
epinid0 ( E ∩ I ) = ∅

Proof of Theorem epinid0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-eprel 5463 . . 3 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
2 df-id 5458 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
31, 2ineq12i 4190 . 2 ( E ∩ I ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
4 inopab 5699 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑦𝑥 = 𝑦)}
5 nelaneq 9055 . . . 4 ¬ (𝑥𝑦𝑥 = 𝑦)
65gen2 1790 . . 3 𝑥𝑦 ¬ (𝑥𝑦𝑥 = 𝑦)
7 opab0 5437 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝑦𝑥 = 𝑦)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥𝑦𝑥 = 𝑦))
86, 7mpbir 232 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑦𝑥 = 𝑦)} = ∅
93, 4, 83eqtri 2852 1 ( E ∩ I ) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wal 1528   = wceq 1530  cin 3938  c0 4294  {copab 5124   I cid 5457   E cep 5462
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 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-reg 9048
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-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-opab 5125  df-id 5458  df-eprel 5463  df-xp 5559  df-rel 5560
This theorem is referenced by: (None)
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