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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 4300 | . 2 ⊢ (𝑋 ∈ dom 𝐹 → dom 𝐹 ≠ ∅) | |
2 | dmeq 5772 | . . . 4 ⊢ (𝐹 = ∅ → dom 𝐹 = dom ∅) | |
3 | dm0 5790 | . . . 4 ⊢ dom ∅ = ∅ | |
4 | 2, 3 | syl6eq 2872 | . . 3 ⊢ (𝐹 = ∅ → dom 𝐹 = ∅) |
5 | 4 | necon3i 3048 | . 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 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 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 |
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-dm 5565 |
This theorem is referenced by: cycpmrn 30785 |
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