Theorem n0snor2el 4793

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 4792	. . . 4 ⊢ (∃𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑤 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
2	1	olcd 885	. . 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 2960	. . . . . . 7 ⊢ (𝑤 ≠ 𝑦 ↔ ¬ 𝑤 = 𝑦)
5	4	rexbii 3111	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑤 = 𝑦)
6		rexnal 3116	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ¬ 𝑤 = 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑤 = 𝑦)
7	5, 6	bitri 277	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑤 = 𝑦)
8	7	ralbii 3110	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ↔ ∀𝑤 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 𝑤 = 𝑦)
9		ralnex 3090	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 𝑤 = 𝑦 ↔ ¬ ∃𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑤 = 𝑦)
10	8, 9	bitri 277	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ↔ ¬ ∃𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑤 = 𝑦)
11		neeq1 3021	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 ≠ 𝑦 ↔ 𝑥 ≠ 𝑦))
12	11	rexbidv 3188	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦))
13	12	rspccva 3582	. . . . . 6 ⊢ ((∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦)
14	13	reximdva0 4310	. . . . 5 ⊢ ((∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦)
15	14	orcd 884	. . . 4 ⊢ ((∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
16	15	ex 416	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑤 ≠ 𝑦 → (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
17	10, 16	sylbir 237	. 2 ⊢ (¬ ∃𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
18	3, 17	pm2.61i 183	1 ⊢ (𝐴 ≠ ∅ → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≠ 𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1562  ∃wex 1801   ≠ wne 2959  ∀wral 3078  ∃wrex 3088  ∅c0 4287  {csn 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-v 3458  df-dif 3909  df-ss 3923  df-nul 4288  df-sn 4585

This theorem is referenced by:  iunopeqop  5492  iunopeqopOLD  5493  1sdom2dom  9200