Theorem pwsnOLD 4824

Description: Obsolete version of pwsn 4823 as of 14-Apr-2024. Note that the proof is essentially the same once one inlines sssn 4752 in the proof of pwsn 4823. (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
pwsnOLD	⊢ 𝒫 {𝐴} = {∅, {𝐴}}

Proof of Theorem pwsnOLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3948	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ {𝐴}))
2		velsn 4576	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
3	2	imbi2i 338	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 → 𝑦 ∈ {𝐴}) ↔ (𝑦 ∈ 𝑥 → 𝑦 = 𝐴))
4	3	albii 1819	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ {𝐴}) ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = 𝐴))
5	1, 4	bitri 277	. . . . . . . 8 ⊢ (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = 𝐴))
6		neq0 4302	. . . . . . . . . 10 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
7		exintr 1892	. . . . . . . . . 10 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = 𝐴) → (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴)))
8	6, 7	syl5bi 244	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = 𝐴) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴)))
9		dfclel 2893	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑥))
10		exancom 1860	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴))
11	9, 10	bitr2i 278	. . . . . . . . . 10 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴) ↔ 𝐴 ∈ 𝑥)
12		snssi 4734	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑥 → {𝐴} ⊆ 𝑥)
13	11, 12	sylbi 219	. . . . . . . . 9 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴) → {𝐴} ⊆ 𝑥)
14	8, 13	syl6 35	. . . . . . . 8 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = 𝐴) → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
15	5, 14	sylbi 219	. . . . . . 7 ⊢ (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
16	15	anc2li 558	. . . . . 6 ⊢ (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥)))
17		eqss 3975	. . . . . 6 ⊢ (𝑥 = {𝐴} ↔ (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥))
18	16, 17	syl6ibr 254	. . . . 5 ⊢ (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → 𝑥 = {𝐴}))
19	18	orrd 859	. . . 4 ⊢ (𝑥 ⊆ {𝐴} → (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
20		0ss 4343	. . . . . 6 ⊢ ∅ ⊆ {𝐴}
21		sseq1 3985	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {𝐴} ↔ ∅ ⊆ {𝐴}))
22	20, 21	mpbiri 260	. . . . 5 ⊢ (𝑥 = ∅ → 𝑥 ⊆ {𝐴})
23		eqimss 4016	. . . . 5 ⊢ (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
24	22, 23	jaoi 853	. . . 4 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
25	19, 24	impbii 211	. . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
26	25	abbii 2885	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
27		df-pw 4534	. 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}}
28		dfpr2 4579	. 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
29	26, 27, 28	3eqtr4i 2853	1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843  ∀wal 1534   = wceq 1536  ∃wex 1779   ∈ wcel 2113  {cab 2798   ⊆ wss 3929  ∅c0 4284  𝒫 cpw 4532  {csn 4560  {cpr 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-pw 4534  df-sn 4561  df-pr 4563

This theorem is referenced by: (None)