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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 5520 | . . 3 ⊢ dom {∅} = ran ◡{∅} | |
2 | dmsn0 5819 | . . 3 ⊢ dom {∅} = ∅ | |
3 | 1, 2 | eqtr3i 2824 | . 2 ⊢ ran ◡{∅} = ∅ |
4 | relcnv 5721 | . . 3 ⊢ Rel ◡{∅} | |
5 | relrn0 5588 | . . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅) |
7 | 3, 6 | mpbir 223 | 1 ⊢ ◡{∅} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1653 ∅c0 4116 {csn 4369 ◡ccnv 5312 dom cdm 5313 ran crn 5314 Rel wrel 5318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pr 5098 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-rab 3099 df-v 3388 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-br 4845 df-opab 4907 df-xp 5319 df-rel 5320 df-cnv 5321 df-dm 5323 df-rn 5324 |
This theorem is referenced by: opswap 5842 brtpos0 7598 tpostpos 7611 |
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