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Theorem mosssn 47936
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 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem mosssn
StepHypRef Expression
1 sssn 4832 . 2 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
2 mo0 47935 . . 3 (𝐴 = ∅ → ∃*𝑥 𝑥𝐴)
3 mosn 47934 . . 3 (𝐴 = {𝐵} → ∃*𝑥 𝑥𝐴)
42, 3jaoi 855 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → ∃*𝑥 𝑥𝐴)
51, 4sylbi 216 1 (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 845   = wceq 1533  wcel 2098  ∃*wmo 2527  wss 3947  c0 4324  {csn 4630
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 2166  ax-ext 2698  ax-sep 5301
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-v 3473  df-sbc 3777  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4325  df-sn 4631
This theorem is referenced by: (None)
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