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Theorem cnvepnep 9646
Description: The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 9644. (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 5589 . . . . . 6 E = {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥}
21cnveqi 5888 . . . . 5 E = {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥}
3 cnvopab 6160 . . . . 5 {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥}
42, 3eqtri 2763 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥}
5 df-eprel 5589 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
64, 5ineq12i 4226 . . 3 ( E ∩ E ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦})
7 inopab 5842 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)}
86, 7eqtri 2763 . 2 ( E ∩ E ) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)}
9 en2lp 9644 . . . 4 ¬ (𝑦𝑥𝑥𝑦)
109gen2 1793 . . 3 𝑥𝑦 ¬ (𝑦𝑥𝑥𝑦)
11 opab0 5564 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑦𝑥𝑥𝑦))
1210, 11mpbir 231 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)} = ∅
138, 12eqtri 2763 1 ( E ∩ E ) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1535   = wceq 1537  cin 3962  c0 4339  {copab 5210   E cep 5588  ccnv 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-fr 5641  df-xp 5695  df-rel 5696  df-cnv 5697
This theorem is referenced by:  epnsym  9647
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