Theorem mosssn 47754

Description: "At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.)

Assertion

Ref	Expression
mosssn	⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mosssn

Step	Hyp	Ref	Expression
1		sssn 4824	. 2 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
2		mo0 47753	. . 3 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
3		mosn 47752	. . 3 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
4	2, 3	jaoi 854	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → ∃*𝑥 𝑥 ∈ 𝐴)
5	1, 4	sylbi 216	1 ⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∨ wo 844   = wceq 1533   ∈ wcel 2098  ∃*wmo 2526   ⊆ wss 3943  ∅c0 4317  {csn 4623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-v 3470  df-sbc 3773  df-dif 3946  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624

This theorem is referenced by: (None)