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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.
StepHypRef Expression
1 relcnv 5649 . 2 Rel
2 rel0 5387 . 2 Rel ∅
3 vex 3331 . . . 4 𝑥 ∈ V
4 vex 3331 . . . 4 𝑦 ∈ V
53, 4opelcnv 5447 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
6 noel 4050 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
7 noel 4050 . . . 4 ¬ ⟨𝑦, 𝑥⟩ ∈ ∅
86, 72false 364 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
95, 8bitr4i 267 . 2 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
101, 2, 9eqrelriiv 5359 1 ∅ = ∅
Colors of variables: wff setvar class
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)
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