Theorem cnveq0b 6191

Description: A class is empty iff it is a relation whose converse is empty. (Contributed by SN, 8-Oct-2026.)

Assertion

Ref	Expression
cnveq0b	⊢ (𝐴 = ∅ ↔ (Rel 𝐴 ∧ ◡𝐴 = ∅))

Proof of Theorem cnveq0b

Step	Hyp	Ref	Expression
1		rel0 5775	. . 3 ⊢ Rel ∅
2		releq 5752	. . 3 ⊢ (𝐴 = ∅ → (Rel 𝐴 ↔ Rel ∅))
3	1, 2	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → Rel 𝐴)
4		cnveq0 6190	. 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))
5	3, 4	biadanii 834	1 ⊢ (𝐴 = ∅ ↔ (Rel 𝐴 ∧ ◡𝐴 = ∅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ∅c0 4278  ^◡ccnv 5649  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-res 5662

This theorem is used by: (None)