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Theorem cnvepnep 9552
Description: The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 9550. (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 5541 . . . . . 6 E = {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥}
21cnveqi 5834 . . . . 5 E = {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥}
3 cnvopab 6095 . . . . 5 {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥}
42, 3eqtri 2761 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥}
5 df-eprel 5541 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
64, 5ineq12i 4174 . . 3 ( E ∩ E ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦})
7 inopab 5789 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)}
86, 7eqtri 2761 . 2 ( E ∩ E ) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)}
9 en2lp 9550 . . . 4 ¬ (𝑦𝑥𝑥𝑦)
109gen2 1799 . . 3 𝑥𝑦 ¬ (𝑦𝑥𝑥𝑦)
11 opab0 5515 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑦𝑥𝑥𝑦))
1210, 11mpbir 230 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)} = ∅
138, 12eqtri 2761 1 ( E ∩ E ) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wal 1540   = wceq 1542  cin 3913  c0 4286  {copab 5171   E cep 5540  ccnv 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-reg 9536
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-eprel 5541  df-fr 5592  df-xp 5643  df-rel 5644  df-cnv 5645
This theorem is referenced by:  epnsym  9553
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