Theorem mosssn 48739

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 4825	. 2 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
2		mo0 48738	. . 3 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
3		mosn 48737	. . 3 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
4	2, 3	jaoi 857	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → ∃*𝑥 𝑥 ∈ 𝐴)
5	1, 4	sylbi 217	1 ⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1539   ∈ wcel 2107  ∃*wmo 2537   ⊆ wss 3950  ∅c0 4332  {csn 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  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-ss 3967  df-nul 4333  df-sn 4626

This theorem is referenced by: (None)