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Theorem eldmne0 31008
Description: A function of nonempty domain is not empty. (Contributed by Thierry Arnoux, 20-Nov-2023.)
Assertion
Ref Expression
eldmne0 (𝑋 ∈ dom 𝐹𝐹 ≠ ∅)

Proof of Theorem eldmne0
StepHypRef Expression
1 ne0i 4274 . 2 (𝑋 ∈ dom 𝐹 → dom 𝐹 ≠ ∅)
2 dmeq 5825 . . . 4 (𝐹 = ∅ → dom 𝐹 = dom ∅)
3 dm0 5842 . . . 4 dom ∅ = ∅
42, 3eqtrdi 2792 . . 3 (𝐹 = ∅ → dom 𝐹 = ∅)
54necon3i 2974 . 2 (dom 𝐹 ≠ ∅ → 𝐹 ≠ ∅)
61, 5syl 17 1 (𝑋 ∈ dom 𝐹𝐹 ≠ ∅)
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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