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Theorem mo0sn 46974
Description: Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 19-Sep-2024.)
Assertion
Ref Expression
mo0sn (∃*𝑥 𝑥𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem mo0sn
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1918 . . 3 𝑧 𝑥𝐴
2 nfv 1918 . . 3 𝑥 𝑧𝐴
3 eleq1w 2821 . . 3 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
41, 2, 3cbvmow 2602 . 2 (∃*𝑥 𝑥𝐴 ↔ ∃*𝑧 𝑧𝐴)
5 neq0 4310 . . . . . . . 8 𝐴 = ∅ ↔ ∃𝑧 𝑧𝐴)
65anbi1i 625 . . . . . . 7 ((¬ 𝐴 = ∅ ∧ ∃*𝑧 𝑧𝐴) ↔ (∃𝑧 𝑧𝐴 ∧ ∃*𝑧 𝑧𝐴))
7 df-eu 2568 . . . . . . 7 (∃!𝑧 𝑧𝐴 ↔ (∃𝑧 𝑧𝐴 ∧ ∃*𝑧 𝑧𝐴))
8 eu6 2573 . . . . . . 7 (∃!𝑧 𝑧𝐴 ↔ ∃𝑦𝑧(𝑧𝐴𝑧 = 𝑦))
96, 7, 83bitr2i 299 . . . . . 6 ((¬ 𝐴 = ∅ ∧ ∃*𝑧 𝑧𝐴) ↔ ∃𝑦𝑧(𝑧𝐴𝑧 = 𝑦))
10 dfcleq 2730 . . . . . . . 8 (𝐴 = {𝑦} ↔ ∀𝑧(𝑧𝐴𝑧 ∈ {𝑦}))
11 velsn 4607 . . . . . . . . . 10 (𝑧 ∈ {𝑦} ↔ 𝑧 = 𝑦)
1211bibi2i 338 . . . . . . . . 9 ((𝑧𝐴𝑧 ∈ {𝑦}) ↔ (𝑧𝐴𝑧 = 𝑦))
1312albii 1822 . . . . . . . 8 (∀𝑧(𝑧𝐴𝑧 ∈ {𝑦}) ↔ ∀𝑧(𝑧𝐴𝑧 = 𝑦))
1410, 13sylbbr 235 . . . . . . 7 (∀𝑧(𝑧𝐴𝑧 = 𝑦) → 𝐴 = {𝑦})
1514eximi 1838 . . . . . 6 (∃𝑦𝑧(𝑧𝐴𝑧 = 𝑦) → ∃𝑦 𝐴 = {𝑦})
169, 15sylbi 216 . . . . 5 ((¬ 𝐴 = ∅ ∧ ∃*𝑧 𝑧𝐴) → ∃𝑦 𝐴 = {𝑦})
1716expcom 415 . . . 4 (∃*𝑧 𝑧𝐴 → (¬ 𝐴 = ∅ → ∃𝑦 𝐴 = {𝑦}))
1817orrd 862 . . 3 (∃*𝑧 𝑧𝐴 → (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
19 mo0 46972 . . . 4 (𝐴 = ∅ → ∃*𝑧 𝑧𝐴)
20 mosn 46971 . . . . 5 (𝐴 = {𝑦} → ∃*𝑧 𝑧𝐴)
2120exlimiv 1934 . . . 4 (∃𝑦 𝐴 = {𝑦} → ∃*𝑧 𝑧𝐴)
2219, 21jaoi 856 . . 3 ((𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}) → ∃*𝑧 𝑧𝐴)
2318, 22impbii 208 . 2 (∃*𝑧 𝑧𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
244, 23bitri 275 1 (∃*𝑥 𝑥𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wo 846  wal 1540   = wceq 1542  wex 1782  wcel 2107  ∃*wmo 2537  ∃!weu 2567  c0 4287  {csn 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-v 3450  df-sbc 3745  df-dif 3918  df-nul 4288  df-sn 4592
This theorem is referenced by:  mosssn2  46975  mofmo  46987  mofeu  46988  f1mo  46993  setc2othin  47150
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