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Theorem cnveq0 5103
Description: A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
cnveq0 (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))

Proof of Theorem cnveq0
StepHypRef Expression
1 cnv0 5050 . 2 ∅ = ∅
2 rel0 4769 . . . . 5 Rel ∅
3 cnveqb 5102 . . . . 5 ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ 𝐴 = ∅))
42, 3mpan2 425 . . . 4 (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))
5 eqeq2 2199 . . . . 5 (∅ = ∅ → (𝐴 = ∅ ↔ 𝐴 = ∅))
65bibi2d 232 . . . 4 (∅ = ∅ → ((𝐴 = ∅ ↔ 𝐴 = ∅) ↔ (𝐴 = ∅ ↔ 𝐴 = ∅)))
74, 6imbitrrid 156 . . 3 (∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅)))
87eqcoms 2192 . 2 (∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅)))
91, 8ax-mp 5 1 (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  c0 3437  ccnv 4643  Rel wrel 4649
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: (None)
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