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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 5624 | . 2 ⊢ ¬ ∅ ∈ (V × V) | |
2 | dmsnn0 6109 | . . 3 ⊢ (∅ ∈ (V × V) ↔ dom {∅} ≠ ∅) | |
3 | 2 | necon2bbii 2997 | . 2 ⊢ (dom {∅} = ∅ ↔ ¬ ∅ ∈ (V × V)) |
4 | 1, 3 | mpbir 230 | 1 ⊢ dom {∅} = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∅c0 4262 {csn 4567 × cxp 5588 dom cdm 5590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ne 2946 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-br 5080 df-opab 5142 df-xp 5596 df-dm 5600 |
This theorem is referenced by: cnvsn0 6112 dmsnopss 6116 1st0 7830 2nd0 7831 |
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