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Theorem eldmne0 31852
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 4335 . 2 (𝑋 ∈ dom 𝐹 → dom 𝐹 ≠ ∅)
2 dmeq 5904 . . . 4 (𝐹 = ∅ → dom 𝐹 = dom ∅)
3 dm0 5921 . . . 4 dom ∅ = ∅
42, 3eqtrdi 2789 . . 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 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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