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Theorem cnvepnep 9065
Description: The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 9063. (Proposed by BJ, 18-Jun-2022.) (Contributed by AV, 19-Jun-2022.)
Assertion
Ref Expression
cnvepnep ( E ∩ E ) = ∅

Proof of Theorem cnvepnep
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-eprel 5464 . . . . . 6 E = {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥}
21cnveqi 5744 . . . . 5 E = {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥}
3 cnvopab 5996 . . . . 5 {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥}
42, 3eqtri 2849 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥}
5 df-eprel 5464 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
64, 5ineq12i 4191 . . 3 ( E ∩ E ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦})
7 inopab 5700 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)}
86, 7eqtri 2849 . 2 ( E ∩ E ) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)}
9 en2lp 9063 . . . 4 ¬ (𝑦𝑥𝑥𝑦)
109gen2 1790 . . 3 𝑥𝑦 ¬ (𝑦𝑥𝑥𝑦)
11 opab0 5438 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑦𝑥𝑥𝑦))
1210, 11mpbir 232 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)} = ∅
138, 12eqtri 2849 1 ( E ∩ E ) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wal 1528   = wceq 1530  cin 3939  c0 4295  {copab 5125   E cep 5463  ccnv 5553
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  ax-reg 9050
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-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  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-eprel 5464  df-fr 5513  df-xp 5560  df-rel 5561  df-cnv 5562
This theorem is referenced by:  epnsym  9066
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