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Theorem mosssn2 48741
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 1992 . 2 (∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}) ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2 sssn 4825 . . 3 (𝐴 ⊆ {𝑦} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝑦}))
32exbii 1847 . 2 (∃𝑦 𝐴 ⊆ {𝑦} ↔ ∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}))
4 mo0sn 48740 . 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 206  wo 847   = wceq 1539  wex 1778  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:  subthinc  49117
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