Theorem cnveq0 6202

Description: A relation is 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		rel0 5781	. . 3 ⊢ Rel ∅
2		cnveqb 6201	. . 3 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))
3	1, 2	mpan2 704	. 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))
4		cnv0 5867	. . 3 ⊢ ◡∅ = ∅
5	4	eqeq2i 2776	. 2 ⊢ (◡𝐴 = ◡∅ ↔ ◡𝐴 = ∅)
6	3, 5	bitrdi 290	1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∅c0 4281  ^◡ccnv 5654  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-res 5667

This theorem is used by:  cnveq0b  6203  elrn3  36704