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Theorem mosssn2 45741
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 2005 . 2 (∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}) ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2 sssn 4724 . . 3 (𝐴 ⊆ {𝑦} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝑦}))
32exbii 1854 . 2 (∃𝑦 𝐴 ⊆ {𝑦} ↔ ∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}))
4 mo0sn 45740 . 2 (∃*𝑥 𝑥𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
51, 3, 43bitr4ri 307 1 (∃*𝑥 𝑥𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 209  wo 846   = wceq 1542  wex 1786  wcel 2114  ∃*wmo 2539  wss 3853  c0 4221  {csn 4526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-v 3402  df-sbc 3686  df-dif 3856  df-in 3860  df-ss 3870  df-nul 4222  df-sn 4527
This theorem is referenced by: (None)
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