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Mirrors > Home > ILE Home > Th. List > dmsn0 | GIF version |
Description: The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.) |
Ref | Expression |
---|---|
dmsn0 | ⊢ dom {∅} = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 4654 | . . . 4 ⊢ ¬ ∅ ∈ (V × V) | |
2 | dmsnm 5094 | . . . 4 ⊢ (∅ ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {∅}) | |
3 | 1, 2 | mtbi 670 | . . 3 ⊢ ¬ ∃𝑥 𝑥 ∈ dom {∅} |
4 | alnex 1499 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom {∅} ↔ ¬ ∃𝑥 𝑥 ∈ dom {∅}) | |
5 | 3, 4 | mpbir 146 | . 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ dom {∅} |
6 | eq0 3441 | . 2 ⊢ (dom {∅} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom {∅}) | |
7 | 5, 6 | mpbir 146 | 1 ⊢ dom {∅} = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1351 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 ∅c0 3422 {csn 3592 × cxp 4624 dom cdm 4626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4004 df-opab 4065 df-xp 4632 df-dm 4636 |
This theorem is referenced by: cnvsn0 5097 1st0 6144 2nd0 6145 |
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