Theorem mosssn2 45778

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 2003	. 2 ⊢ (∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}) ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2		sssn 4725	. . 3 ⊢ (𝐴 ⊆ {𝑦} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝑦}))
3	2	exbii 1855	. 2 ⊢ (∃𝑦 𝐴 ⊆ {𝑦} ↔ ∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}))
4		mo0sn 45777	. 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 847   = wceq 1543  ∃wex 1787   ∈ wcel 2112  ∃*wmo 2537   ⊆ wss 3853  ∅c0 4223  {csn 4527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-v 3400  df-sbc 3684  df-dif 3856  df-in 3860  df-ss 3870  df-nul 4224  df-sn 4528

This theorem is referenced by:  subthinc  45937