Theorem mosssn2 48665

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 1991	. 2 ⊢ (∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}) ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
2		sssn 4831	. . 3 ⊢ (𝐴 ⊆ {𝑦} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝑦}))
3	2	exbii 1845	. 2 ⊢ (∃𝑦 𝐴 ⊆ {𝑦} ↔ ∃𝑦(𝐴 = ∅ ∨ 𝐴 = {𝑦}))
4		mo0sn 48664	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
5	1, 3, 4	3bitr4ri 304	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})

Syntax hints:   ↔ wb 206   ∨ wo 847   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∃*wmo 2536   ⊆ wss 3963  ∅c0 4339  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-v 3480  df-sbc 3792  df-dif 3966  df-ss 3980  df-nul 4340  df-sn 4632

This theorem is referenced by:  subthinc  48840