Theorem cnveq0 5185

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

Step	Hyp	Ref	Expression
1		cnv0 5132	. 2 ⊢ ◡∅ = ∅
2		rel0 4844	. . . . 5 ⊢ Rel ∅
3		cnveqb 5184	. . . . 5 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))
4	2, 3	mpan2 425	. . . 4 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))
5		eqeq2 2239	. . . . 5 ⊢ (∅ = ◡∅ → (◡𝐴 = ∅ ↔ ◡𝐴 = ◡∅))
6	5	bibi2d 232	. . . 4 ⊢ (∅ = ◡∅ → ((𝐴 = ∅ ↔ ◡𝐴 = ∅) ↔ (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)))
7	4, 6	imbitrrid 156	. . 3 ⊢ (∅ = ◡∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)))
8	7	eqcoms 2232	. 2 ⊢ (◡∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)))
9	1, 8	ax-mp 5	1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  ∅c0 3491  ^◡ccnv 4718  Rel wrel 4724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727

This theorem is referenced by: (None)