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Theorem n0snor2el 4756
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 4755 . . . 4 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
21olcd 870 . . 3 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
32a1d 25 . 2 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
4 df-ne 3014 . . . . . . 7 (𝑤𝑦 ↔ ¬ 𝑤 = 𝑦)
54rexbii 3244 . . . . . 6 (∃𝑦𝐴 𝑤𝑦 ↔ ∃𝑦𝐴 ¬ 𝑤 = 𝑦)
6 rexnal 3235 . . . . . 6 (∃𝑦𝐴 ¬ 𝑤 = 𝑦 ↔ ¬ ∀𝑦𝐴 𝑤 = 𝑦)
75, 6bitri 276 . . . . 5 (∃𝑦𝐴 𝑤𝑦 ↔ ¬ ∀𝑦𝐴 𝑤 = 𝑦)
87ralbii 3162 . . . 4 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 ↔ ∀𝑤𝐴 ¬ ∀𝑦𝐴 𝑤 = 𝑦)
9 ralnex 3233 . . . 4 (∀𝑤𝐴 ¬ ∀𝑦𝐴 𝑤 = 𝑦 ↔ ¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦)
108, 9bitri 276 . . 3 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 ↔ ¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦)
11 neeq1 3075 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
1211rexbidv 3294 . . . . . . 7 (𝑤 = 𝑥 → (∃𝑦𝐴 𝑤𝑦 ↔ ∃𝑦𝐴 𝑥𝑦))
1312rspccva 3619 . . . . . 6 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝑥𝐴) → ∃𝑦𝐴 𝑥𝑦)
1413reximdva0 4309 . . . . 5 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
1514orcd 869 . . . 4 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝐴 ≠ ∅) → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
1615ex 413 . . 3 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
1710, 16sylbir 236 . 2 (¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
183, 17pm2.61i 183 1 (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wex 1771  wne 3013  wral 3135  wrex 3136  c0 4288  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-v 3494  df-dif 3936  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558
This theorem is referenced by:  iunopeqop  5402
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