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Mirrors > Home > MPE Home > Th. List > dmsn0el | Structured version Visualization version GIF version |
Description: The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.) |
Ref | Expression |
---|---|
dmsn0el | ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmsnn0 6064 | . . 3 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
2 | 0nelelxp 5590 | . . 3 ⊢ (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴) | |
3 | 1, 2 | sylbir 237 | . 2 ⊢ (dom {𝐴} ≠ ∅ → ¬ ∅ ∈ 𝐴) |
4 | 3 | necon4ai 3047 | 1 ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∅c0 4291 {csn 4567 × cxp 5553 dom cdm 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-dm 5565 |
This theorem is referenced by: dmsnsnsn 6077 |
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