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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 6160 | . 2 ⊢ ◡∅ = ∅ | |
| 2 | rel0 5809 | . . . . 5 ⊢ Rel ∅ | |
| 3 | cnveqb 6216 | . . . . 5 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | |
| 4 | 2, 3 | mpan2 691 | . . . 4 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | 
| 5 | eqeq2 2749 | . . . . 5 ⊢ (∅ = ◡∅ → (◡𝐴 = ∅ ↔ ◡𝐴 = ◡∅)) | |
| 6 | 5 | bibi2d 342 | . . . 4 ⊢ (∅ = ◡∅ → ((𝐴 = ∅ ↔ ◡𝐴 = ∅) ↔ (𝐴 = ∅ ↔ ◡𝐴 = ◡∅))) | 
| 7 | 4, 6 | imbitrrid 246 | . . 3 ⊢ (∅ = ◡∅ → (Rel 𝐴 → (𝐴 = ∅ ↔ ◡𝐴 = ∅))) | 
| 8 | 7 | eqcoms 2745 | . 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 206 = wceq 1540 ∅c0 4333 ◡ccnv 5684 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 | 
| This theorem is referenced by: elrn3 35762 | 
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