Theorem eldmne0 32705

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 4293	. 2 ⊢ (𝑋 ∈ dom 𝐹 → dom 𝐹 ≠ ∅)
2		dmeq 5852	. . . 4 ⊢ (𝐹 = ∅ → dom 𝐹 = dom ∅)
3		dm0 5869	. . . 4 ⊢ dom ∅ = ∅
4	2, 3	eqtrdi 2787	. . 3 ⊢ (𝐹 = ∅ → dom 𝐹 = ∅)
5	4	necon3i 2964	. 2 ⊢ (dom 𝐹 ≠ ∅ → 𝐹 ≠ ∅)
6	1, 5	syl 17	1 ⊢ (𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∅c0 4285  dom cdm 5624

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-dm 5634

This theorem is referenced by:  cycpmrn  33225