Theorem cnveq0 5060

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 5007	. 2 ⊢ ◡∅ = ∅
2		rel0 4729	. . . . 5 ⊢ Rel ∅
3		cnveqb 5059	. . . . 5 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))
4	2, 3	mpan2 422	. . . 4 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))
5		eqeq2 2175	. . . . 5 ⊢ (∅ = ◡∅ → (◡𝐴 = ∅ ↔ ◡𝐴 = ◡∅))
6	5	bibi2d 231	. . . 4 ⊢ (∅ = ◡∅ → ((𝐴 = ∅ ↔ ◡𝐴 = ∅) ↔ (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)))
7	4, 6	syl5ibr 155	. . 3 ⊢ (∅ = ◡∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)))
8	7	eqcoms 2168	. 2 ⊢ (◡∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)))
9	1, 8	ax-mp 5	1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  ∅c0 3409  ^◡ccnv 4603  Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612

This theorem is referenced by: (None)