Theorem mo0 48706

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 48704	. . . 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 48705	. 2 ⊢ (𝐴 = {V} → ∃*𝑥 𝑥 ∈ 𝐴)
6	4, 5	syl 17	1 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∃*wmo 2537  Vcvv 3479  ∅c0 4332  {csn 4624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-v 3481  df-sbc 3788  df-dif 3953  df-nul 4333  df-sn 4625

This theorem is referenced by:  mosssn  48707  mo0sn  48708  f1omo  48765