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Theorem epinid0 9640
Description: The membership relation and the identity relation are disjoint. Variable-free version of nelaneq 9639. (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 5584 . . 3 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
2 df-id 5578 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
31, 2ineq12i 4218 . 2 ( E ∩ I ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦})
4 inopab 5839 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑦𝑥 = 𝑦)}
5 nelaneq 9639 . . . 4 ¬ (𝑥𝑦𝑥 = 𝑦)
65gen2 1796 . . 3 𝑥𝑦 ¬ (𝑥𝑦𝑥 = 𝑦)
7 opab0 5559 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝑦𝑥 = 𝑦)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑥𝑦𝑥 = 𝑦))
86, 7mpbir 231 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑦𝑥 = 𝑦)} = ∅
93, 4, 83eqtri 2769 1 ( E ∩ I ) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1538   = wceq 1540  cin 3950  c0 4333  {copab 5205   I cid 5577   E cep 5583
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-reg 9632
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-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  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-opab 5206  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692
This theorem is referenced by: (None)
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