Theorem cnvepnep 9602

Description: The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 9600. (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.

Step	Hyp	Ref	Expression
1		df-eprel 5580	. . . . . 6 ⊢ E = {⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥}
2	1	cnveqi 5874	. . . . 5 ⊢ ◡ E = ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥}
3		cnvopab 6138	. . . . 5 ⊢ ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥}
4	2, 3	eqtri 2760	. . . 4 ⊢ ◡ E = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥}
5		df-eprel 5580	. . . 4 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
6	4, 5	ineq12i 4210	. . 3 ⊢ (◡ E ∩ E ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦})
7		inopab 5829	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)}
8	6, 7	eqtri 2760	. 2 ⊢ (◡ E ∩ E ) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)}
9		en2lp 9600	. . . 4 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
10	9	gen2 1798	. . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
11		opab0 5554	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦))
12	10, 11	mpbir 230	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅
13	8, 12	eqtri 2760	1 ⊢ (◡ E ∩ E ) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 396  ∀wal 1539   = wceq 1541   ∩ cin 3947  ∅c0 4322  {copab 5210   E cep 5579  ^◡ccnv 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-reg 9586

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-eprel 5580  df-fr 5631  df-xp 5682  df-rel 5683  df-cnv 5684

This theorem is referenced by:  epnsym  9603