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Mirrors > Home > MPE Home > Th. List > cnveq0 | Structured version Visualization version GIF version |
Description: A relation empty iff its converse is empty. (Contributed by FL, 19-Sep-2011.) |
Ref | Expression |
---|---|
cnveq0 | ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnv0 6093 | . 2 ⊢ ◡∅ = ∅ | |
2 | rel0 5755 | . . . . 5 ⊢ Rel ∅ | |
3 | cnveqb 6148 | . . . . 5 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | |
4 | 2, 3 | mpan2 689 | . . . 4 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) |
5 | eqeq2 2748 | . . . . 5 ⊢ (∅ = ◡∅ → (◡𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | |
6 | 5 | bibi2d 342 | . . . 4 ⊢ (∅ = ◡∅ → ((𝐴 = ∅ ↔ ◡𝐴 = ∅) ↔ (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))) |
7 | 4, 6 | syl5ibr 245 | . . 3 ⊢ (∅ = ◡∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))) |
8 | 7 | eqcoms 2744 | . 2 ⊢ (◡∅ = ∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))) |
9 | 1, 8 | ax-mp 5 | 1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∅c0 4282 ◡ccnv 5632 Rel wrel 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-xp 5639 df-rel 5640 df-cnv 5641 |
This theorem is referenced by: elrn3 34335 |
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