Theorem cnv0OLD 5455

Description: Obsolete version of cnv0 5454 as of 25-Oct-2021. (Contributed by NM, 6-Apr-1998.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cnv0OLD	⊢ ◡∅ = ∅

Proof of Theorem cnv0OLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 5422	. 2 ⊢ Rel ◡∅
2		rel0 5166	. 2 ⊢ Rel ∅
3		vex 3176	. . . 4 ⊢ 𝑥 ∈ V
4		vex 3176	. . . 4 ⊢ 𝑦 ∈ V
5	3, 4	opelcnv 5226	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
6		noel 3878	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
7		noel 3878	. . . 4 ⊢ ¬ ⟨𝑦, 𝑥⟩ ∈ ∅
8	6, 7	2false 364	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
9	5, 8	bitr4i 266	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∅ ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
10	1, 2, 9	eqrelriiv 5137	1 ⊢ ◡∅ = ∅

Syntax hints:   = wceq 1475   ∈ wcel 1977  ∅c0 3874  ⟨cop 4131  ^◡ccnv 5037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046

This theorem is referenced by: (None)