Theorem n0snor2el 4858

Description: A nonempty set is either a singleton or contains at least two different elements. (Contributed by AV, 20-Sep-2020.)

Assertion

Ref	Expression
n0snor2el	⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))

Distinct variable groups:   𝑦,𝐴,𝑥   𝑧,𝐴

Proof of Theorem n0snor2el

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issn 4857	. . . 4 ⊢ (∃𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑤 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
2	1	olcd 873	. . 3 ⊢ (∃𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑤 = 𝑦 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
3	2	a1d 25	. 2 ⊢ (∃𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
4		df-ne 2947	. . . . . . 7 ⊢ (𝑤 ≠ 𝑦 ↔ ¬ 𝑤 = 𝑦)
5	4	rexbii 3100	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑤 = 𝑦)
6		rexnal 3106	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑤 = 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑤 = 𝑦)
7	5, 6	bitri 275	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑤 = 𝑦)
8	7	ralbii 3099	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ↔ ∀𝑤 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 𝑤 = 𝑦)
9		ralnex 3078	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 𝑤 = 𝑦 ↔ ¬ ∃𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑤 = 𝑦)
10	8, 9	bitri 275	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ↔ ¬ ∃𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑤 = 𝑦)
11		neeq1 3009	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 ≠ 𝑦 ↔ 𝑥 ≠ 𝑦))
12	11	rexbidv 3185	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦))
13	12	rspccva 3634	. . . . . 6 ⊢ ((∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦)
14	13	reximdva0 4378	. . . . 5 ⊢ ((∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦)
15	14	orcd 872	. . . 4 ⊢ ((∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
16	15	ex 412	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 → (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
17	10, 16	sylbir 235	. 2 ⊢ (¬ ∃𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
18	3, 17	pm2.61i 182	1 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537  ∃wex 1777   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  ∅c0 4352  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353  df-sn 4649

This theorem is referenced by:  iunopeqop  5540  1sdom2dom  9310