Theorem eldmne0 32611

Description: A function of nonempty domain is not empty. (Contributed by Thierry Arnoux, 20-Nov-2023.)

Assertion

Ref	Expression
eldmne0	⊢ (𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅)

Proof of Theorem eldmne0

Step	Hyp	Ref	Expression
1		ne0i 4321	. 2 ⊢ (𝑋 ∈ dom 𝐹 → dom 𝐹 ≠ ∅)
2		dmeq 5888	. . . 4 ⊢ (𝐹 = ∅ → dom 𝐹 = dom ∅)
3		dm0 5905	. . . 4 ⊢ dom ∅ = ∅
4	2, 3	eqtrdi 2787	. . 3 ⊢ (𝐹 = ∅ → dom 𝐹 = ∅)
5	4	necon3i 2965	. 2 ⊢ (dom 𝐹 ≠ ∅ → 𝐹 ≠ ∅)
6	1, 5	syl 17	1 ⊢ (𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∅c0 4313  dom cdm 5659

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-dm 5669

This theorem is referenced by:  cycpmrn  33159