Theorem cnvepnep 9529

Description: The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 9527. (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 5532	. . . . . 6 ⊢ E = {⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥}
2	1	cnveqi 5831	. . . . 5 ⊢ ◡ E = ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥}
3		cnvopab 6102	. . . . 5 ⊢ ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥}
4	2, 3	eqtri 2760	. . . 4 ⊢ ◡ E = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥}
5		df-eprel 5532	. . . 4 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
6	4, 5	ineq12i 4172	. . 3 ⊢ (◡ E ∩ E ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦})
7		inopab 5786	. . 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 9527	. . . 4 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
10	9	gen2 1798	. . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
11		opab0 5510	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦))
12	10, 11	mpbir 231	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)} = ∅
13	8, 12	eqtri 2760	1 ⊢ (◡ E ∩ E ) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1540   = wceq 1542   ∩ cin 3902  ∅c0 4287  {copab 5162   E cep 5531  ^◡ccnv 5631

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-reg 9509

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-eprel 5532  df-fr 5585  df-xp 5638  df-rel 5639  df-cnv 5640

This theorem is referenced by:  epnsym  9530