Theorem cnvepnep 9498

Description: The membership (epsilon) relation and its converse are disjoint, i.e., E is an asymmetric relation. Variable-free version of en2lp 9496. (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 5514	. . . . . 6 ⊢ E = {⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥}
2	1	cnveqi 5813	. . . . 5 ⊢ ◡ E = ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥}
3		cnvopab 6083	. . . . 5 ⊢ ◡{⟨𝑦, 𝑥⟩ ∣ 𝑦 ∈ 𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥}
4	2, 3	eqtri 2754	. . . 4 ⊢ ◡ E = {⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥}
5		df-eprel 5514	. . . 4 ⊢ E = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}
6	4, 5	ineq12i 4165	. . 3 ⊢ (◡ E ∩ E ) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦})
7		inopab 5768	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦 ∈ 𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)}
8	6, 7	eqtri 2754	. 2 ⊢ (◡ E ∩ E ) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)}
9		en2lp 9496	. . . 4 ⊢ ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
10	9	gen2 1797	. . 3 ⊢ ∀𝑥∀𝑦 ¬ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦)
11		opab0 5492	. . 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 2754	1 ⊢ (◡ E ∩ E ) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1539   = wceq 1541   ∩ cin 3896  ∅c0 4280  {copab 5151   E cep 5513  ^◡ccnv 5613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-reg 9478

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-eprel 5514  df-fr 5567  df-xp 5620  df-rel 5621  df-cnv 5622

This theorem is referenced by:  epnsym  9499