Theorem mo0 50256

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 50254	. . . 4 ⊢ {V} = ∅
2	1	eqcomi 2772	. . 3 ⊢ ∅ = {V}
3		eqeq1 2767	. . 3 ⊢ (𝐴 = ∅ → (𝐴 = {V} ↔ ∅ = {V}))
4	2, 3	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → 𝐴 = {V})
5		mosn 50255	. 2 ⊢ (𝐴 = {V} → ∃*𝑥 𝑥 ∈ 𝐴)
6	4, 5	syl 18	1 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∃*wmo 2565  Vcvv 3454  ∅c0 4281  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-v 3456  df-sbc 3743  df-dif 4050  df-nul 4282  df-sn 4589

This theorem is used by:  mosssn  50257  mo0sn  50258  f1omo  50338  f1omoOLD  50339  discthing  50927