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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 4274 | . 2 ⊢ (𝑋 ∈ dom 𝐹 → dom 𝐹 ≠ ∅) | |
2 | dmeq 5825 | . . . 4 ⊢ (𝐹 = ∅ → dom 𝐹 = dom ∅) | |
3 | dm0 5842 | . . . 4 ⊢ dom ∅ = ∅ | |
4 | 2, 3 | eqtrdi 2792 | . . 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 1539 ∈ wcel 2104 ≠ wne 2941 ∅c0 4262 dom cdm 5600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-br 5082 df-dm 5610 |
This theorem is referenced by: cycpmrn 31455 |
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