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Theorem mosssn2 46162
Description: Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.)
Assertion
Ref Expression
mosssn2 (∃*𝑥 𝑥𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem mosssn2
StepHypRef Expression
1 19.45v 1997 . 2 (∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}) ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2 sssn 4759 . . 3 (𝐴 ⊆ {𝑦} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝑦}))
32exbii 1850 . 2 (∃𝑦 𝐴 ⊆ {𝑦} ↔ ∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}))
4 mo0sn 46161 . 2 (∃*𝑥 𝑥𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
51, 3, 43bitr4ri 304 1 (∃*𝑥 𝑥𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 844   = wceq 1539  wex 1782  wcel 2106  ∃*wmo 2538  wss 3887  c0 4256  {csn 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-v 3434  df-sbc 3717  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562
This theorem is referenced by:  subthinc  46321
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