Theorem mo0 46159

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 46157	. . . 4 ⊢ {V} = ∅
2	1	eqcomi 2747	. . 3 ⊢ ∅ = {V}
3		eqeq1 2742	. . 3 ⊢ (𝐴 = ∅ → (𝐴 = {V} ↔ ∅ = {V}))
4	2, 3	mpbiri 257	. 2 ⊢ (𝐴 = ∅ → 𝐴 = {V})
5		mosn 46158	. 2 ⊢ (𝐴 = {V} → ∃*𝑥 𝑥 ∈ 𝐴)
6	4, 5	syl 17	1 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∃*wmo 2538  Vcvv 3432  ∅c0 4256  {csn 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-v 3434  df-sbc 3717  df-dif 3890  df-nul 4257  df-sn 4562

This theorem is referenced by:  mosssn  46160  mo0sn  46161  f1omo  46188