Theorem mosssn2 49315

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 2006	. 2 ⊢ (∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}) ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2		sssn 4758	. . 3 ⊢ (𝐴 ⊆ {𝑦} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝑦}))
3	2	exbii 1855	. 2 ⊢ (∃𝑦 𝐴 ⊆ {𝑦} ↔ ∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}))
4		mo0sn 49314	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
5	1, 3, 4	3bitr4ri 305	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})

Syntax hints:   ↔ wb 207   ∨ wo 853   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃*wmo 2541   ⊆ wss 3883  ∅c0 4262  {csn 4556

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-v 3433  df-sbc 3724  df-dif 3886  df-ss 3900  df-nul 4263  df-sn 4557

This theorem is referenced by:  subthinc  49941