Theorem mosssn 49477

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 4796	. 2 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
2		mo0 49476	. . 3 ⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
3		mosn 49475	. . 3 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
4	2, 3	jaoi 870	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → ∃*𝑥 𝑥 ∈ 𝐴)
5	1, 4	sylbi 220	1 ⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1567   ∈ wcel 2149  ∃*wmo 2571   ⊆ wss 3913  ∅c0 4294  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-v 3465  df-sbc 3754  df-dif 3916  df-ss 3930  df-nul 4295  df-sn 4595

This theorem is referenced by: (None)