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Theorem cnv0 5050
Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.)
Assertion
Ref Expression
cnv0 ∅ = ∅

Proof of Theorem cnv0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5024 . 2 Rel
2 rel0 4769 . 2 Rel ∅
3 vex 2755 . . . 4 𝑥 ∈ V
4 vex 2755 . . . 4 𝑦 ∈ V
53, 4opelcnv 4827 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
6 noel 3441 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
7 noel 3441 . . . 4 ¬ ⟨𝑦, 𝑥⟩ ∈ ∅
86, 72false 702 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
95, 8bitr4i 187 . 2 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
101, 2, 9eqrelriiv 4738 1 ∅ = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wcel 2160  c0 3437  cop 3610  ccnv 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652
This theorem is referenced by:  xp0  5066  cnveq0  5103  co01  5161  f10  5514  f1o00  5515  tpos0  6300
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