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Theorem cnv0 4900
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 4875 . 2 Rel
2 rel0 4624 . 2 Rel ∅
3 vex 2660 . . . 4 𝑥 ∈ V
4 vex 2660 . . . 4 𝑦 ∈ V
53, 4opelcnv 4681 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
6 noel 3333 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
7 noel 3333 . . . 4 ¬ ⟨𝑦, 𝑥⟩ ∈ ∅
86, 72false 673 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
95, 8bitr4i 186 . 2 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
101, 2, 9eqrelriiv 4593 1 ∅ = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1314  wcel 1463  c0 3329  cop 3496  ccnv 4498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-cnv 4507
This theorem is referenced by:  xp0  4916  cnveq0  4953  co01  5011  f10  5357  f1o00  5358  tpos0  6125
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