Theorem elpsseu 36592

Description: A class has a unique element just in case it has a unique proper subset. (Contributed by Adrian Ducourtial, 26-Oct-2026.)

Assertion

Ref	Expression
elpsseu	⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑦 𝑦 ⊊ 𝐴)

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem elpsseu

Step	Hyp	Ref	Expression
1		elpssex 36590	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ⊊ 𝐴)
2		elpssmo 36591	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑦 𝑦 ⊊ 𝐴)
3	1, 2	anbi12i 640	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴) ↔ (∃𝑦 𝑦 ⊊ 𝐴 ∧ ∃*𝑦 𝑦 ⊊ 𝐴))
4		df-eu 2597	. 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴))
5		df-eu 2597	. 2 ⊢ (∃!𝑦 𝑦 ⊊ 𝐴 ↔ (∃𝑦 𝑦 ⊊ 𝐴 ∧ ∃*𝑦 𝑦 ⊊ 𝐴))
6	3, 4, 5	3bitr4i 306	1 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑦 𝑦 ⊊ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1811   ∈ wcel 2146  ∃*wmo 2565  ∃!weu 2596   ⊊ wpss 4016

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591

This theorem is used by: (None)