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Theorem cnv0OLD 5495
Description: Obsolete version of cnv0 5494 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 5462 . 2 Rel
2 rel0 5204 . 2 Rel ∅
3 vex 3189 . . . 4 𝑥 ∈ V
4 vex 3189 . . . 4 𝑦 ∈ V
53, 4opelcnv 5264 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
6 noel 3895 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
7 noel 3895 . . . 4 ¬ ⟨𝑦, 𝑥⟩ ∈ ∅
86, 72false 365 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
95, 8bitr4i 267 . 2 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
101, 2, 9eqrelriiv 5175 1 ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  c0 3891  cop 4154  ccnv 5073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082
This theorem is referenced by: (None)
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