Theorem eldmne0 31250

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 4281	. 2 ⊢ (𝑋 ∈ dom 𝐹 → dom 𝐹 ≠ ∅)
2		dmeq 5845	. . . 4 ⊢ (𝐹 = ∅ → dom 𝐹 = dom ∅)
3		dm0 5862	. . . 4 ⊢ dom ∅ = ∅
4	2, 3	eqtrdi 2792	. . 3 ⊢ (𝐹 = ∅ → dom 𝐹 = ∅)
5	4	necon3i 2973	. 2 ⊢ (dom 𝐹 ≠ ∅ → 𝐹 ≠ ∅)
6	1, 5	syl 17	1 ⊢ (𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105   ≠ wne 2940  ∅c0 4269  dom cdm 5620

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-dm 5630

This theorem is referenced by:  cycpmrn  31697