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Theorem n0snor2el 4744
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 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
Distinct variable groups:   𝑦,𝐴,𝑥   𝑧,𝐴

Proof of Theorem n0snor2el
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 issn 4743 . . . 4 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
21olcd 874 . . 3 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
32a1d 25 . 2 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
4 df-ne 2941 . . . . . . 7 (𝑤𝑦 ↔ ¬ 𝑤 = 𝑦)
54rexbii 3170 . . . . . 6 (∃𝑦𝐴 𝑤𝑦 ↔ ∃𝑦𝐴 ¬ 𝑤 = 𝑦)
6 rexnal 3160 . . . . . 6 (∃𝑦𝐴 ¬ 𝑤 = 𝑦 ↔ ¬ ∀𝑦𝐴 𝑤 = 𝑦)
75, 6bitri 278 . . . . 5 (∃𝑦𝐴 𝑤𝑦 ↔ ¬ ∀𝑦𝐴 𝑤 = 𝑦)
87ralbii 3088 . . . 4 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 ↔ ∀𝑤𝐴 ¬ ∀𝑦𝐴 𝑤 = 𝑦)
9 ralnex 3158 . . . 4 (∀𝑤𝐴 ¬ ∀𝑦𝐴 𝑤 = 𝑦 ↔ ¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦)
108, 9bitri 278 . . 3 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 ↔ ¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦)
11 neeq1 3003 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
1211rexbidv 3216 . . . . . . 7 (𝑤 = 𝑥 → (∃𝑦𝐴 𝑤𝑦 ↔ ∃𝑦𝐴 𝑥𝑦))
1312rspccva 3536 . . . . . 6 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝑥𝐴) → ∃𝑦𝐴 𝑥𝑦)
1413reximdva0 4266 . . . . 5 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
1514orcd 873 . . . 4 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝐴 ≠ ∅) → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
1615ex 416 . . 3 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
1710, 16sylbir 238 . 2 (¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
183, 17pm2.61i 185 1 (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 847   = wceq 1543  wex 1787  wne 2940  wral 3061  wrex 3062  c0 4237  {csn 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-v 3410  df-dif 3869  df-in 3873  df-ss 3883  df-nul 4238  df-sn 4542
This theorem is referenced by:  iunopeqop  5404
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