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Theorem cnv0 4754
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 4730 . 2 Rel
2 rel0 4489 . 2 Rel ∅
3 vex 2577 . . . 4 𝑥 ∈ V
4 vex 2577 . . . 4 𝑦 ∈ V
53, 4opelcnv 4544 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
6 noel 3255 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
7 noel 3255 . . . 4 ¬ ⟨𝑦, 𝑥⟩ ∈ ∅
86, 72false 627 . . 3 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑦, 𝑥⟩ ∈ ∅)
95, 8bitr4i 180 . 2 (⟨𝑥, 𝑦⟩ ∈ ∅ ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)
101, 2, 9eqrelriiv 4461 1 ∅ = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  c0 3251  cop 3405  ccnv 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380
This theorem is referenced by:  xp0  4770  cnveq0  4804  co01  4862  f10  5187  f1o00  5188  tpos0  5919
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