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Theorem epinid0 9052
Description: The membership (epsilon) relation and the identity relation are disjoint. Variable-free version of nelaneq 9051. (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 5458 . . 3 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
2 df-id 5453 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
31, 2ineq12i 4184 . 2 ( E ∩ I ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
4 inopab 5694 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑦𝑥 = 𝑦)}
5 nelaneq 9051 . . . 4 ¬ (𝑥𝑦𝑥 = 𝑦)
65gen2 1788 . . 3 𝑥𝑦 ¬ (𝑥𝑦𝑥 = 𝑦)
7 opab0 5432 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝑦𝑥 = 𝑦)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥𝑦𝑥 = 𝑦))
86, 7mpbir 232 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑦𝑥 = 𝑦)} = ∅
93, 4, 83eqtri 2845 1 ( E ∩ I ) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wal 1526   = wceq 1528  cin 3932  c0 4288  {copab 5119   I cid 5452   E cep 5457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-reg 9044
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5120  df-id 5453  df-eprel 5458  df-xp 5554  df-rel 5555
This theorem is referenced by: (None)
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