Theorem mo0sn 45777

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 1922	. . 3 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
2		nfv 1922	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3		eleq1w 2813	. . 3 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
4	1, 2, 3	cbvmow 2602	. 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑧 𝑧 ∈ 𝐴)
5		neq0 4246	. . . . . . . 8 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴)
6	5	anbi1i 627	. . . . . . 7 ⊢ ((¬ 𝐴 = ∅ ∧ ∃*𝑧 𝑧 ∈ 𝐴) ↔ (∃𝑧 𝑧 ∈ 𝐴 ∧ ∃*𝑧 𝑧 ∈ 𝐴))
7		df-eu 2568	. . . . . . 7 ⊢ (∃!𝑧 𝑧 ∈ 𝐴 ↔ (∃𝑧 𝑧 ∈ 𝐴 ∧ ∃*𝑧 𝑧 ∈ 𝐴))
8		eu6 2573	. . . . . . 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 2729	. . . . . . . 8 ⊢ (𝐴 = {𝑦} ↔ ∀𝑧(𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑦}))
11		velsn 4543	. . . . . . . . . 10 ⊢ (𝑧 ∈ {𝑦} ↔ 𝑧 = 𝑦)
12	11	bibi2i 341	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ↔ 𝑧 ∈ {𝑦}) ↔ (𝑧 ∈ 𝐴 ↔ 𝑧 = 𝑦))
13	12	albii 1827	. . . . . . . 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 1842	. . . . . 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 417	. . . 4 ⊢ (∃*𝑧 𝑧 ∈ 𝐴 → (¬ 𝐴 = ∅ → ∃𝑦 𝐴 = {𝑦}))
18	17	orrd 863	. . 3 ⊢ (∃*𝑧 𝑧 ∈ 𝐴 → (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
19		mo0 45775	. . . 4 ⊢ (𝐴 = ∅ → ∃*𝑧 𝑧 ∈ 𝐴)
20		mosn 45774	. . . . 5 ⊢ (𝐴 = {𝑦} → ∃*𝑧 𝑧 ∈ 𝐴)
21	20	exlimiv 1938	. . . 4 ⊢ (∃𝑦 𝐴 = {𝑦} → ∃*𝑧 𝑧 ∈ 𝐴)
22	19, 21	jaoi 857	. . 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 399   ∨ wo 847  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2112  ∃*wmo 2537  ∃!weu 2567  ∅c0 4223  {csn 4527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-v 3400  df-sbc 3684  df-dif 3856  df-nul 4224  df-sn 4528

This theorem is referenced by:  mosssn2  45778  mofmo  45790  mofeu  45791  f1mo  45796  setc2othin  45953