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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldmne0 | Structured version Visualization version GIF version |
Description: A function of nonempty domain is not empty. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
eldmne0 | ⊢ (𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4335 | . 2 ⊢ (𝑋 ∈ dom 𝐹 → dom 𝐹 ≠ ∅) | |
2 | dmeq 5904 | . . . 4 ⊢ (𝐹 = ∅ → dom 𝐹 = dom ∅) | |
3 | dm0 5921 | . . . 4 ⊢ dom ∅ = ∅ | |
4 | 2, 3 | eqtrdi 2789 | . . 3 ⊢ (𝐹 = ∅ → dom 𝐹 = ∅) |
5 | 4 | necon3i 2974 | . 2 ⊢ (dom 𝐹 ≠ ∅ → 𝐹 ≠ ∅) |
6 | 1, 5 | syl 17 | 1 ⊢ (𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∅c0 4323 dom cdm 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-dm 5687 |
This theorem is referenced by: cycpmrn 32302 |
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