Theorem mo0 49001

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 48999	. . . 4 ⊢ {V} = ∅
2	1	eqcomi 2743	. . 3 ⊢ ∅ = {V}
3		eqeq1 2738	. . 3 ⊢ (𝐴 = ∅ → (𝐴 = {V} ↔ ∅ = {V}))
4	2, 3	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → 𝐴 = {V})
5		mosn 49000	. 2 ⊢ (𝐴 = {V} → ∃*𝑥 𝑥 ∈ 𝐴)
6	4, 5	syl 17	1 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∃*wmo 2535  Vcvv 3438  ∅c0 4283  {csn 4578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-v 3440  df-sbc 3739  df-dif 3902  df-nul 4284  df-sn 4579

This theorem is referenced by:  mosssn  49002  mo0sn  49003  f1omo  49080  f1omoOLD  49081  discthing  49648