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Theorem n0snor2el 4727
 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 group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem n0snor2el
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 issn 4726 . . . 4 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
21olcd 871 . . 3 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
32a1d 25 . 2 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
4 df-ne 2988 . . . . . . 7 (𝑤𝑦 ↔ ¬ 𝑤 = 𝑦)
54rexbii 3211 . . . . . 6 (∃𝑦𝐴 𝑤𝑦 ↔ ∃𝑦𝐴 ¬ 𝑤 = 𝑦)
6 rexnal 3201 . . . . . 6 (∃𝑦𝐴 ¬ 𝑤 = 𝑦 ↔ ¬ ∀𝑦𝐴 𝑤 = 𝑦)
75, 6bitri 278 . . . . 5 (∃𝑦𝐴 𝑤𝑦 ↔ ¬ ∀𝑦𝐴 𝑤 = 𝑦)
87ralbii 3133 . . . 4 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 ↔ ∀𝑤𝐴 ¬ ∀𝑦𝐴 𝑤 = 𝑦)
9 ralnex 3199 . . . 4 (∀𝑤𝐴 ¬ ∀𝑦𝐴 𝑤 = 𝑦 ↔ ¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦)
108, 9bitri 278 . . 3 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 ↔ ¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦)
11 neeq1 3049 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
1211rexbidv 3257 . . . . . . 7 (𝑤 = 𝑥 → (∃𝑦𝐴 𝑤𝑦 ↔ ∃𝑦𝐴 𝑥𝑦))
1312rspccva 3571 . . . . . 6 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝑥𝐴) → ∃𝑦𝐴 𝑥𝑦)
1413reximdva0 4268 . . . . 5 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
1514orcd 870 . . . 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 844   = wceq 1538  ∃wex 1781   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∅c0 4246  {csn 4528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3444  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4247  df-sn 4529 This theorem is referenced by:  iunopeqop  5380
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