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Theorem cnveq0 6147
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 6091 . 2 ∅ = ∅
2 rel0 5753 . . . . 5 Rel ∅
3 cnveqb 6146 . . . . 5 ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ 𝐴 = ∅))
42, 3mpan2 689 . . . 4 (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))
5 eqeq2 2748 . . . . 5 (∅ = ∅ → (𝐴 = ∅ ↔ 𝐴 = ∅))
65bibi2d 342 . . . 4 (∅ = ∅ → ((𝐴 = ∅ ↔ 𝐴 = ∅) ↔ (𝐴 = ∅ ↔ 𝐴 = ∅)))
74, 6syl5ibr 245 . . 3 (∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅)))
87eqcoms 2744 . 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 205   = wceq 1541  c0 4280  ccnv 5630  Rel wrel 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-rel 5638  df-cnv 5639
This theorem is referenced by:  elrn3  34167
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