MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnveq0 Structured version   Visualization version   GIF version

Theorem cnveq0 5846
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 5792 . 2 ∅ = ∅
2 rel0 5472 . . . . 5 Rel ∅
3 cnveqb 5845 . . . . 5 ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ 𝐴 = ∅))
42, 3mpan2 681 . . . 4 (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅))
5 eqeq2 2789 . . . . 5 (∅ = ∅ → (𝐴 = ∅ ↔ 𝐴 = ∅))
65bibi2d 334 . . . 4 (∅ = ∅ → ((𝐴 = ∅ ↔ 𝐴 = ∅) ↔ (𝐴 = ∅ ↔ 𝐴 = ∅)))
74, 6syl5ibr 238 . . 3 (∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ 𝐴 = ∅)))
87eqcoms 2786 . 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 198   = wceq 1601  c0 4141  ccnv 5356  Rel wrel 5362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365
This theorem is referenced by:  elrn3  32254
  Copyright terms: Public domain W3C validator