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Theorem cnveq0 6219
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 6163 . 2 ∅ = ∅
2 rel0 5812 . . . . 5 Rel ∅
3 cnveqb 6218 . . . . 5 ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ 𝐴 = ∅))
42, 3mpan2 691 . . . 4 (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))
5 eqeq2 2747 . . . . 5 (∅ = ∅ → (𝐴 = ∅ ↔ 𝐴 = ∅))
65bibi2d 342 . . . 4 (∅ = ∅ → ((𝐴 = ∅ ↔ 𝐴 = ∅) ↔ (𝐴 = ∅ ↔ 𝐴 = ∅)))
74, 6imbitrrid 246 . . 3 (∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅)))
87eqcoms 2743 . 2 (∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅)))
91, 8ax-mp 5 1 (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  c0 4339  ccnv 5688  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697
This theorem is referenced by:  elrn3  35742
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