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Theorem eldmne0 30391
 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 4253 . 2 (𝑋 ∈ dom 𝐹 → dom 𝐹 ≠ ∅)
2 dmeq 5740 . . . 4 (𝐹 = ∅ → dom 𝐹 = dom ∅)
3 dm0 5758 . . . 4 dom ∅ = ∅
42, 3eqtrdi 2852 . . 3 (𝐹 = ∅ → dom 𝐹 = ∅)
54necon3i 3022 . 2 (dom 𝐹 ≠ ∅ → 𝐹 ≠ ∅)
61, 5syl 17 1 (𝑋 ∈ dom 𝐹𝐹 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∅c0 4246  dom cdm 5523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-dm 5533 This theorem is referenced by:  cycpmrn  30839
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