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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 4575 | . . . 4 ⊢ ¬ ∅ ∈ (V × V) | |
2 | dmsnm 5012 | . . . 4 ⊢ (∅ ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {∅}) | |
3 | 1, 2 | mtbi 660 | . . 3 ⊢ ¬ ∃𝑥 𝑥 ∈ dom {∅} |
4 | alnex 1476 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom {∅} ↔ ¬ ∃𝑥 𝑥 ∈ dom {∅}) | |
5 | 3, 4 | mpbir 145 | . 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ dom {∅} |
6 | eq0 3386 | . 2 ⊢ (dom {∅} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom {∅}) | |
7 | 5, 6 | mpbir 145 | 1 ⊢ dom {∅} = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1330 = wceq 1332 ∃wex 1469 ∈ wcel 1481 Vcvv 2689 ∅c0 3368 {csn 3532 × cxp 4545 dom cdm 4547 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-xp 4553 df-dm 4557 |
This theorem is referenced by: cnvsn0 5015 1st0 6050 2nd0 6051 |
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