Theorem eldmne0 32953

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 4295	. 2 ⊢ (𝑋 ∈ dom 𝐹 → dom 𝐹 ≠ ∅)
2		dmeq 5895	. . . 4 ⊢ (𝐹 = ∅ → dom 𝐹 = dom ∅)
3		dm0 5912	. . . 4 ⊢ dom ∅ = ∅
4	2, 3	eqtrdi 2814	. . 3 ⊢ (𝐹 = ∅ → dom 𝐹 = ∅)
5	4	necon3i 2990	. 2 ⊢ (dom 𝐹 ≠ ∅ → 𝐹 ≠ ∅)
6	1, 5	syl 18	1 ⊢ (𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∅c0 4287  dom cdm 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-dm 5673

This theorem is referenced by:  cycpmrn  33444