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Mirrors > Home > MPE Home > Th. List > dmsn0 | Structured version Visualization version 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 5553 | . 2 ⊢ ¬ ∅ ∈ (V × V) | |
2 | dmsnn0 6031 | . . 3 ⊢ (∅ ∈ (V × V) ↔ dom {∅} ≠ ∅) | |
3 | 2 | necon2bbii 3038 | . 2 ⊢ (dom {∅} = ∅ ↔ ¬ ∅ ∈ (V × V)) |
4 | 1, 3 | mpbir 234 | 1 ⊢ dom {∅} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {csn 4525 × cxp 5517 dom cdm 5519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-dm 5529 |
This theorem is referenced by: cnvsn0 6034 dmsnopss 6038 1st0 7677 2nd0 7678 |
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