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| Mirrors > Home > MPE Home > Th. List > cnvsn0 | Structured version Visualization version GIF version | ||
| Description: The converse of the singleton of the empty set is empty. (Contributed by Mario Carneiro, 30-Aug-2015.) |
| Ref | Expression |
|---|---|
| cnvsn0 | ⊢ ◡{∅} = ∅ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfdm4 5906 | . . 3 ⊢ dom {∅} = ran ◡{∅} | |
| 2 | dmsn0 6229 | . . 3 ⊢ dom {∅} = ∅ | |
| 3 | 1, 2 | eqtr3i 2767 | . 2 ⊢ ran ◡{∅} = ∅ |
| 4 | relcnv 6122 | . . 3 ⊢ Rel ◡{∅} | |
| 5 | relrn0 5983 | . . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)) | |
| 6 | 4, 5 | ax-mp 5 | . 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅) |
| 7 | 3, 6 | mpbir 231 | 1 ⊢ ◡{∅} = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∅c0 4333 {csn 4626 ◡ccnv 5684 dom cdm 5685 ran crn 5686 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-10 2141 ax-11 2157 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-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 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 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: opswap 6249 brtpos0 8258 tpostpos 8271 dftpos5 48774 |
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