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Theorem n0snor2el 4339
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 4338 . . . 4 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → ∃𝑧 𝐴 = {𝑧})
21olcd 408 . . 3 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
32a1d 25 . 2 (∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦 → (𝐴 ≠ ∅ → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧})))
4 df-ne 2791 . . . . . . 7 (𝑤𝑦 ↔ ¬ 𝑤 = 𝑦)
54rexbii 3036 . . . . . 6 (∃𝑦𝐴 𝑤𝑦 ↔ ∃𝑦𝐴 ¬ 𝑤 = 𝑦)
6 rexnal 2991 . . . . . 6 (∃𝑦𝐴 ¬ 𝑤 = 𝑦 ↔ ¬ ∀𝑦𝐴 𝑤 = 𝑦)
75, 6bitri 264 . . . . 5 (∃𝑦𝐴 𝑤𝑦 ↔ ¬ ∀𝑦𝐴 𝑤 = 𝑦)
87ralbii 2976 . . . 4 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 ↔ ∀𝑤𝐴 ¬ ∀𝑦𝐴 𝑤 = 𝑦)
9 ralnex 2988 . . . 4 (∀𝑤𝐴 ¬ ∀𝑦𝐴 𝑤 = 𝑦 ↔ ¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦)
108, 9bitri 264 . . 3 (∀𝑤𝐴𝑦𝐴 𝑤𝑦 ↔ ¬ ∃𝑤𝐴𝑦𝐴 𝑤 = 𝑦)
11 neeq1 2852 . . . . . . . 8 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
1211rexbidv 3047 . . . . . . 7 (𝑤 = 𝑥 → (∃𝑦𝐴 𝑤𝑦 ↔ ∃𝑦𝐴 𝑥𝑦))
1312rspccva 3298 . . . . . 6 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝑥𝐴) → ∃𝑦𝐴 𝑥𝑦)
1413reximdva0 3915 . . . . 5 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝐴 ≠ ∅) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
1514orcd 407 . . . 4 ((∀𝑤𝐴𝑦𝐴 𝑤𝑦𝐴 ≠ ∅) → (∃𝑥𝐴𝑦𝐴 𝑥𝑦 ∨ ∃𝑧 𝐴 = {𝑧}))
1615ex 450 . . 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  wo 383  wa 384   = wceq 1480  wex 1701  wne 2790  wral 2908  wrex 2909  c0 3897  {csn 4155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-v 3192  df-dif 3563  df-in 3567  df-ss 3574  df-nul 3898  df-sn 4156
This theorem is referenced by:  iunopeqop  4951
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