Theorem mosssn2 49514

Description: Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.)

Assertion

Ref	Expression
mosssn2	⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem mosssn2

Step	Hyp	Ref	Expression
1		19.45v 2026	. 2 ⊢ (∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}) ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2		sssn 4796	. . 3 ⊢ (𝐴 ⊆ {𝑦} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝑦}))
3	2	exbii 1875	. 2 ⊢ (∃𝑦 𝐴 ⊆ {𝑦} ↔ ∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}))
4		mo0sn 49513	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
5	1, 3, 4	3bitr4ri 307	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})

Syntax hints:   ↔ wb 209   ∨ wo 860   = wceq 1567  ∃wex 1806   ∈ 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:  subthinc  50140