Theorem cnv0OLD 5682

Description: Obsolete version of cnv0 5681 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 5649	. 2 ⊢ Rel ◡∅
2		rel0 5387	. 2 ⊢ Rel ∅
3		vex 3331	. . . 4 ⊢ 𝑥 ∈ V
4		vex 3331	. . . 4 ⊢ 𝑦 ∈ V
5	3, 4	opelcnv 5447	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
6		noel 4050	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
7		noel 4050	. . . 4 ⊢ ¬ ⟨𝑦, 𝑥⟩ ∈ ∅
8	6, 7	2false 364	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
9	5, 8	bitr4i 267	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡∅ ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
10	1, 2, 9	eqrelriiv 5359	1 ⊢ ◡∅ = ∅

Syntax hints:   = wceq 1620   ∈ wcel 2127  ∅c0 4046  ⟨cop 4315  ^◡ccnv 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260  df-rel 5261  df-cnv 5262

This theorem is referenced by: (None)