Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnveq0 Structured version   GIF version

Theorem cnveq0 4720
 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 4670 . 2 ∅ = ∅
2 rel0 4405 . . . . 5 Rel ∅
3 cnveqb 4719 . . . . 5 ((Rel A Rel ∅) → (A = ∅ ↔ A = ∅))
42, 3mpan2 401 . . . 4 (Rel A → (A = ∅ ↔ A = ∅))
5 eqeq2 2046 . . . . 5 (∅ = ∅ → (A = ∅ ↔ A = ∅))
65bibi2d 221 . . . 4 (∅ = ∅ → ((A = ∅ ↔ A = ∅) ↔ (A = ∅ ↔ A = ∅)))
74, 6syl5ibr 145 . . 3 (∅ = ∅ → (Rel A → (A = ∅ ↔ A = ∅)))
87eqcoms 2040 . 2 (∅ = ∅ → (Rel A → (A = ∅ ↔ A = ∅)))
91, 8ax-mp 7 1 (Rel A → (A = ∅ ↔ A = ∅))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∅c0 3218  ◡ccnv 4287  Rel wrel 4293 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator