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Theorem cnv0 4949
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 4924 . 2 Rel
2 rel0 4671 . 2 Rel ∅
3 vex 2692 . . . 4 𝑥 ∈ V
4 vex 2692 . . . 4 𝑦 ∈ V
53, 4opelcnv 4728 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
6 noel 3371 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
7 noel 3371 . . . 4 ¬ ⟨𝑦, 𝑥⟩ ∈ ∅
86, 72false 691 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
95, 8bitr4i 186 . 2 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
101, 2, 9eqrelriiv 4640 1 ∅ = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1332  wcel 1481  c0 3367  cop 3534  ccnv 4545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553  df-cnv 4554
This theorem is referenced by:  xp0  4965  cnveq0  5002  co01  5060  f10  5408  f1o00  5409  tpos0  6178
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