Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6050 | . . 3 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅) | |
2 | 0nelelxp 5571 | . . 3 ⊢ (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴) | |
3 | 1, 2 | sylbir 238 | . 2 ⊢ (dom {𝐴} ≠ ∅ → ¬ ∅ ∈ 𝐴) |
4 | 3 | necon4ai 2963 | 1 ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 Vcvv 3398 ∅c0 4223 {csn 4527 × cxp 5534 dom cdm 5536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-dm 5546 |
This theorem is referenced by: dmsnsnsn 6063 |
Copyright terms: Public domain | W3C validator |