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Mirrors > Home > ILE Home > Th. List > cnvsn0 | 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 4545 | . . 3 ⊢ dom {∅} = ran ◡{∅} | |
2 | dmsn0 4808 | . . 3 ⊢ dom {∅} = ∅ | |
3 | 1, 2 | eqtr3i 2103 | . 2 ⊢ ran ◡{∅} = ∅ |
4 | relcnv 4723 | . . 3 ⊢ Rel ◡{∅} | |
5 | relrn0 4612 | . . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)) | |
6 | 4, 5 | ax-mp 7 | . 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅) |
7 | 3, 6 | mpbir 144 | 1 ⊢ ◡{∅} = ∅ |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 = wceq 1284 ∅c0 3251 {csn 3398 ◡ccnv 4362 dom cdm 4363 ran crn 4364 Rel wrel 4368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 |
This theorem is referenced by: brtpos0 5890 tpostpos 5902 |
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