Theorem cnvepnep 9648

Description: The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 9646. (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 5584	. . . . . 6 ⊢ E = {⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥}
2	1	cnveqi 5885	. . . . 5 ⊢ ◡ E = ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥}
3		cnvopab 6157	. . . . 5 ⊢ ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥}
4	2, 3	eqtri 2765	. . . 4 ⊢ ◡ E = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥}
5		df-eprel 5584	. . . 4 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
6	4, 5	ineq12i 4218	. . 3 ⊢ (◡ E ∩ E ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦})
7		inopab 5839	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)}
8	6, 7	eqtri 2765	. 2 ⊢ (◡ E ∩ E ) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)}
9		en2lp 9646	. . . 4 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
10	9	gen2 1796	. . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
11		opab0 5559	. . 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 2765	1 ⊢ (◡ E ∩ E ) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1538   = wceq 1540   ∩ cin 3950  ∅c0 4333  {copab 5205   E cep 5583  ^◡ccnv 5684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-reg 9632

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-fr 5637  df-xp 5691  df-rel 5692  df-cnv 5693

This theorem is referenced by:  epnsym  9649