Theorem mo0sn 47712

Description: Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 19-Sep-2024.)

Assertion

Ref	Expression
mo0sn	⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem mo0sn

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1909	. . 3 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
2		nfv 1909	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3		eleq1w 2808	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
4	1, 2, 3	cbvmow 2589	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑧 𝑧 ∈ 𝐴)
5		neq0 4338	. . . . . . . 8 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
6	5	anbi1i 623	. . . . . . 7 ⊢ ((¬ 𝐴 = ∅ ∧ ∃*𝑧 𝑧 ∈ 𝐴) ↔ (∃𝑧 𝑧 ∈ 𝐴 ∧ ∃*𝑧 𝑧 ∈ 𝐴))
7		df-eu 2555	. . . . . . 7 ⊢ (∃!𝑧 𝑧 ∈ 𝐴 ↔ (∃𝑧 𝑧 ∈ 𝐴 ∧ ∃*𝑧 𝑧 ∈ 𝐴))
8		eu6 2560	. . . . . . 7 ⊢ (∃!𝑧 𝑧 ∈ 𝐴 ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦))
9	6, 7, 8	3bitr2i 299	. . . . . 6 ⊢ ((¬ 𝐴 = ∅ ∧ ∃*𝑧 𝑧 ∈ 𝐴) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦))
10		dfcleq 2717	. . . . . . . 8 ⊢ (𝐴 = {𝑦} ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑦}))
11		velsn 4637	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑦} ↔ 𝑧 = 𝑦)
12	11	bibi2i 337	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑦}) ↔ (𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦))
13	12	albii 1813	. . . . . . . 8 ⊢ (∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑦}) ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦))
14	10, 13	sylbbr 235	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦) → 𝐴 = {𝑦})
15	14	eximi 1829	. . . . . 6 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦) → ∃𝑦 𝐴 = {𝑦})
16	9, 15	sylbi 216	. . . . 5 ⊢ ((¬ 𝐴 = ∅ ∧ ∃*𝑧 𝑧 ∈ 𝐴) → ∃𝑦 𝐴 = {𝑦})
17	16	expcom 413	. . . 4 ⊢ (∃*𝑧 𝑧 ∈ 𝐴 → (¬ 𝐴 = ∅ → ∃𝑦 𝐴 = {𝑦}))
18	17	orrd 860	. . 3 ⊢ (∃*𝑧 𝑧 ∈ 𝐴 → (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
19		mo0 47710	. . . 4 ⊢ (𝐴 = ∅ → ∃*𝑧 𝑧 ∈ 𝐴)
20		mosn 47709	. . . . 5 ⊢ (𝐴 = {𝑦} → ∃*𝑧 𝑧 ∈ 𝐴)
21	20	exlimiv 1925	. . . 4 ⊢ (∃𝑦 𝐴 = {𝑦} → ∃*𝑧 𝑧 ∈ 𝐴)
22	19, 21	jaoi 854	. . 3 ⊢ ((𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}) → ∃*𝑧 𝑧 ∈ 𝐴)
23	18, 22	impbii 208	. 2 ⊢ (∃*𝑧 𝑧 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
24	4, 23	bitri 275	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∨ wo 844  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃*wmo 2524  ∃!weu 2554  ∅c0 4315  {csn 4621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-v 3468  df-sbc 3771  df-dif 3944  df-nul 4316  df-sn 4622

This theorem is referenced by:  mosssn2  47713  mofmo  47725  mofeu  47726  f1mo  47731  setc2othin  47888