MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvepnep Structured version   Visualization version   GIF version

Theorem cnvepnep 8751
Description: The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 8750. (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 5223 . . . . . 6 E = {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥}
21cnveqi 5498 . . . . 5 E = {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥}
3 cnvopab 5749 . . . . 5 {⟨𝑦, 𝑥⟩ ∣ 𝑦𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥}
42, 3eqtri 2819 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥}
5 df-eprel 5223 . . . 4 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
64, 5ineq12i 4008 . . 3 ( E ∩ E ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦})
7 inopab 5454 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)}
86, 7eqtri 2819 . 2 ( E ∩ E ) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)}
9 en2lp 8750 . . . 4 ¬ (𝑦𝑥𝑥𝑦)
109gen2 1892 . . 3 𝑥𝑦 ¬ (𝑦𝑥𝑥𝑦)
11 opab0 5201 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)} = ∅ ↔ ∀𝑥𝑦 ¬ (𝑦𝑥𝑥𝑦))
1210, 11mpbir 223 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑦𝑥𝑥𝑦)} = ∅
138, 12eqtri 2819 1 ( E ∩ E ) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 385  wal 1651   = wceq 1653  cin 3766  c0 4113  {copab 4903   E cep 5222  ccnv 5309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-reg 8737
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-eprel 5223  df-fr 5269  df-xp 5316  df-rel 5317  df-cnv 5318
This theorem is referenced by:  epnsym  8752
  Copyright terms: Public domain W3C validator