Theorem eldmne0 32779

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 5877	. . . 4 ⊢ (𝐹 = ∅ → dom 𝐹 = dom ∅)
3		dm0 5894	. . . 4 ⊢ dom ∅ = ∅
4	2, 3	eqtrdi 2812	. . 3 ⊢ (𝐹 = ∅ → dom 𝐹 = ∅)
5	4	necon3i 2988	. 2 ⊢ (dom 𝐹 ≠ ∅ → 𝐹 ≠ ∅)
6	1, 5	syl 17	1 ⊢ (𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∅c0 4285  dom cdm 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-dm 5655

This theorem is referenced by:  cycpmrn  33284