Theorem mo0 49289

Description: "At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
mo0	⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem mo0

Step	Hyp	Ref	Expression
1		vsn 49287	. . . 4 ⊢ {V} = ∅
2	1	eqcomi 2745	. . 3 ⊢ ∅ = {V}
3		eqeq1 2740	. . 3 ⊢ (𝐴 = ∅ → (𝐴 = {V} ↔ ∅ = {V}))
4	2, 3	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → 𝐴 = {V})
5		mosn 49288	. 2 ⊢ (𝐴 = {V} → ∃*𝑥 𝑥 ∈ 𝐴)
6	4, 5	syl 17	1 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∃*wmo 2537  Vcvv 3429  ∅c0 4273  {csn 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-v 3431  df-sbc 3729  df-dif 3892  df-nul 4274  df-sn 4568

This theorem is referenced by:  mosssn  49290  mo0sn  49291  f1omo  49368  f1omoOLD  49369  discthing  49936