Theorem mosssn 49433

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 4784	. 2 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
2		mo0 49432	. . 3 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
3		mosn 49431	. . 3 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
4	2, 3	jaoi 868	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → ∃*𝑥 𝑥 ∈ 𝐴)
5	1, 4	sylbi 219	1 ⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∨ wo 858   = wceq 1560   ∈ wcel 2142  ∃*wmo 2564   ⊆ wss 3904  ∅c0 4285  {csn 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-v 3456  df-sbc 3745  df-dif 3907  df-ss 3921  df-nul 4286  df-sn 4583

This theorem is referenced by: (None)