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Theorem n0snor2el 4497
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 4496 . . . 4 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
21olcd 863 . . 3 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
32a1d 25 . 2 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
4 df-ne 2944 . . . . . . 7 (𝑤𝑦 ↔ ¬ 𝑤 = 𝑦)
54rexbii 3189 . . . . . 6 (∃𝑦𝐴 𝑤𝑦 ↔ ∃𝑦𝐴 ¬ 𝑤 = 𝑦)
6 rexnal 3143 . . . . . 6 (∃𝑦𝐴 ¬ 𝑤 = 𝑦 ↔ ¬ ∀𝑦𝐴 𝑤 = 𝑦)
75, 6bitri 264 . . . . 5 (∃𝑦𝐴 𝑤𝑦 ↔ ¬ ∀𝑦𝐴 𝑤 = 𝑦)
87ralbii 3129 . . . 4 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 ↔ ∀𝑤𝐴 ¬ ∀𝑦𝐴 𝑤 = 𝑦)
9 ralnex 3141 . . . 4 (∀𝑤𝐴 ¬ ∀𝑦𝐴 𝑤 = 𝑦 ↔ ¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦)
108, 9bitri 264 . . 3 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 ↔ ¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦)
11 neeq1 3005 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
1211rexbidv 3200 . . . . . . 7 (𝑤 = 𝑥 → (∃𝑦𝐴 𝑤𝑦 ↔ ∃𝑦𝐴 𝑥𝑦))
1312rspccva 3459 . . . . . 6 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝑥𝐴) → ∃𝑦𝐴 𝑥𝑦)
1413reximdva0 4080 . . . . 5 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
1514orcd 862 . . . 4 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝐴 ≠ ∅) → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
1615ex 397 . . 3 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
1710, 16sylbir 225 . 2 (¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
183, 17pm2.61i 176 1 (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wo 836   = wceq 1631  wex 1852  wne 2943  wral 3061  wrex 3062  c0 4063  {csn 4316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-nul 4064  df-sn 4317
This theorem is referenced by:  iunopeqop  5114
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