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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 5758 | . . 3 ⊢ dom {∅} = ran ◡{∅} | |
2 | dmsn0 6060 | . . 3 ⊢ dom {∅} = ∅ | |
3 | 1, 2 | eqtr3i 2846 | . 2 ⊢ ran ◡{∅} = ∅ |
4 | relcnv 5961 | . . 3 ⊢ Rel ◡{∅} | |
5 | relrn0 5834 | . . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅) |
7 | 3, 6 | mpbir 233 | 1 ⊢ ◡{∅} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∅c0 4290 {csn 4560 ◡ccnv 5548 dom cdm 5549 ran crn 5550 Rel wrel 5554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 |
This theorem is referenced by: opswap 6080 brtpos0 7893 tpostpos 7906 |
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