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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 4731 | . . 3 ⊢ dom {∅} = ran ◡{∅} | |
2 | dmsn0 5006 | . . 3 ⊢ dom {∅} = ∅ | |
3 | 1, 2 | eqtr3i 2162 | . 2 ⊢ ran ◡{∅} = ∅ |
4 | relcnv 4917 | . . 3 ⊢ Rel ◡{∅} | |
5 | relrn0 4801 | . . 3 ⊢ (Rel ◡{∅} → (◡{∅} = ∅ ↔ ran ◡{∅} = ∅)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ (◡{∅} = ∅ ↔ ran ◡{∅} = ∅) |
7 | 3, 6 | mpbir 145 | 1 ⊢ ◡{∅} = ∅ |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∅c0 3363 {csn 3527 ◡ccnv 4538 dom cdm 4539 ran crn 4540 Rel wrel 4544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: brtpos0 6149 tpostpos 6161 |
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