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Theorem cnveq0 5626
 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 5570 . 2 ∅ = ∅
2 rel0 5276 . . . . 5 Rel ∅
3 cnveqb 5625 . . . . 5 ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ 𝐴 = ∅))
42, 3mpan2 707 . . . 4 (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))
5 eqeq2 2662 . . . . 5 (∅ = ∅ → (𝐴 = ∅ ↔ 𝐴 = ∅))
65bibi2d 331 . . . 4 (∅ = ∅ → ((𝐴 = ∅ ↔ 𝐴 = ∅) ↔ (𝐴 = ∅ ↔ 𝐴 = ∅)))
74, 6syl5ibr 236 . . 3 (∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅)))
87eqcoms 2659 . 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 196   = wceq 1523  ∅c0 3948  ◡ccnv 5142  Rel wrel 5148 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151 This theorem is referenced by:  elrn3  31778
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