Theorem mo0sn 49478

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 1941	. . 3 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
2		nfv 1941	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3		eleq1w 2852	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
4	1, 2, 3	cbvmow 2637	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑧 𝑧 ∈ 𝐴)
5		neq0 4314	. . . . . . . 8 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
6	5	anbi1i 635	. . . . . . 7 ⊢ ((¬ 𝐴 = ∅ ∧ ∃*𝑧 𝑧 ∈ 𝐴) ↔ (∃𝑧 𝑧 ∈ 𝐴 ∧ ∃*𝑧 𝑧 ∈ 𝐴))
7		df-eu 2603	. . . . . . 7 ⊢ (∃!𝑧 𝑧 ∈ 𝐴 ↔ (∃𝑧 𝑧 ∈ 𝐴 ∧ ∃*𝑧 𝑧 ∈ 𝐴))
8		eu6 2608	. . . . . . 7 ⊢ (∃!𝑧 𝑧 ∈ 𝐴 ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦))
9	6, 7, 8	3bitr2i 302	. . . . . 6 ⊢ ((¬ 𝐴 = ∅ ∧ ∃*𝑧 𝑧 ∈ 𝐴) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦))
10		dfcleq 2762	. . . . . . . 8 ⊢ (𝐴 = {𝑦} ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑦}))
11		velsn 4610	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑦} ↔ 𝑧 = 𝑦)
12	11	bibi2i 340	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑦}) ↔ (𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦))
13	12	albii 1846	. . . . . . . 8 ⊢ (∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑦}) ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦))
14	10, 13	sylbbr 239	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦) → 𝐴 = {𝑦})
15	14	eximi 1862	. . . . . 6 ⊢ (∃𝑦∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦) → ∃𝑦 𝐴 = {𝑦})
16	9, 15	sylbi 220	. . . . 5 ⊢ ((¬ 𝐴 = ∅ ∧ ∃*𝑧 𝑧 ∈ 𝐴) → ∃𝑦 𝐴 = {𝑦})
17	16	expcom 418	. . . 4 ⊢ (∃*𝑧 𝑧 ∈ 𝐴 → (¬ 𝐴 = ∅ → ∃𝑦 𝐴 = {𝑦}))
18	17	orrd 876	. . 3 ⊢ (∃*𝑧 𝑧 ∈ 𝐴 → (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
19		mo0 49476	. . . 4 ⊢ (𝐴 = ∅ → ∃*𝑧 𝑧 ∈ 𝐴)
20		mosn 49475	. . . . 5 ⊢ (𝐴 = {𝑦} → ∃*𝑧 𝑧 ∈ 𝐴)
21	20	exlimiv 1957	. . . 4 ⊢ (∃𝑦 𝐴 = {𝑦} → ∃*𝑧 𝑧 ∈ 𝐴)
22	19, 21	jaoi 870	. . 3 ⊢ ((𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}) → ∃*𝑧 𝑧 ∈ 𝐴)
23	18, 22	impbii 212	. 2 ⊢ (∃*𝑧 𝑧 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
24	4, 23	bitri 278	1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∨ wo 860  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃*wmo 2571  ∃!weu 2602  ∅c0 4294  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-v 3465  df-sbc 3754  df-dif 3916  df-nul 4295  df-sn 4595

This theorem is referenced by:  mosssn2  49479  mofmo  49509  mofeu  49510  f1mo  49515  setc2othin  50128