Theorem elpssex 36590

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

Assertion

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

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem elpssex

Step	Hyp	Ref	Expression
1		0ss 4353	. . . . . 6 ⊢ ∅ ⊆ 𝐴
2	1	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∅ ⊆ 𝐴)
3		ne0i 4290	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
4	3	necomd 3014	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ∅ ≠ 𝐴)
5		df-pss 4017	. . . . 5 ⊢ (∅ ⊊ 𝐴 ↔ (∅ ⊆ 𝐴 ∧ ∅ ≠ 𝐴))
6	2, 4, 5	sylanbrc 595	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∅ ⊊ 𝐴)
7		0ex 5264	. . . . 5 ⊢ ∅ ∈ V
8		psseq1 4020	. . . . 5 ⊢ (𝑦 = ∅ → (𝑦 ⊊ 𝐴 ↔ ∅ ⊊ 𝐴))
9	7, 8	spcev 3564	. . . 4 ⊢ (∅ ⊊ 𝐴 → ∃𝑦 𝑦 ⊊ 𝐴)
10	6, 9	syl 18	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ⊊ 𝐴)
11	10	exlimiv 1964	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑦 𝑦 ⊊ 𝐴)
12		pssnel 4427	. . . 4 ⊢ (𝑦 ⊊ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝑦))
13		exsimpl 1900	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝑦) → ∃𝑥 𝑥 ∈ 𝐴)
14	12, 13	syl 18	. . 3 ⊢ (𝑦 ⊊ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
15	14	exlimiv 1964	. 2 ⊢ (∃𝑦 𝑦 ⊊ 𝐴 → ∃𝑥 𝑥 ∈ 𝐴)
16	11, 15	impbii 212	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ⊊ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401  ∃wex 1811   ∈ wcel 2146   ≠ wne 2959   ⊆ wss 3900   ⊊ wpss 4016  ∅c0 4281

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-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  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-nul 4282

This theorem is used by:  elpsseu  36592