Theorem pwsnOLD 4900

Description: Obsolete version of pwsn 4899 as of 14-Apr-2024. Note that the proof is essentially the same once one inlines sssn 4828 in the proof of pwsn 4899. (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 3967	. . . . . . . . 9 ⊢ (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ {𝐴}))
2		velsn 4643	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝐴} ↔ 𝑦 = 𝐴)
3	2	imbi2i 335	. . . . . . . . . 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 274	. . . . . . . 8 ⊢ (𝑥 ⊆ {𝐴} ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = 𝐴))
6		neq0 4344	. . . . . . . . . 10 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
7		exintr 1893	. . . . . . . . . 10 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = 𝐴) → (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴)))
8	6, 7	biimtrid 241	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 = 𝐴) → (¬ 𝑥 = ∅ → ∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴)))
9		dfclel 2809	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑥))
10		exancom 1862	. . . . . . . . . . 11 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝑥) ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴))
11	9, 10	bitr2i 275	. . . . . . . . . 10 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ 𝑦 = 𝐴) ↔ 𝐴 ∈ 𝑥)
12		snssi 4810	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑥 → {𝐴} ⊆ 𝑥)
13	11, 12	sylbi 216	. . . . . . . . 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 216	. . . . . . 7 ⊢ (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → {𝐴} ⊆ 𝑥))
16	15	anc2li 554	. . . . . 6 ⊢ (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥)))
17		eqss 3996	. . . . . 6 ⊢ (𝑥 = {𝐴} ↔ (𝑥 ⊆ {𝐴} ∧ {𝐴} ⊆ 𝑥))
18	16, 17	imbitrrdi 251	. . . . 5 ⊢ (𝑥 ⊆ {𝐴} → (¬ 𝑥 = ∅ → 𝑥 = {𝐴}))
19	18	orrd 859	. . . 4 ⊢ (𝑥 ⊆ {𝐴} → (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
20		0ss 4395	. . . . . 6 ⊢ ∅ ⊆ {𝐴}
21		sseq1 4006	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ {𝐴} ↔ ∅ ⊆ {𝐴}))
22	20, 21	mpbiri 257	. . . . 5 ⊢ (𝑥 = ∅ → 𝑥 ⊆ {𝐴})
23		eqimss 4039	. . . . 5 ⊢ (𝑥 = {𝐴} → 𝑥 ⊆ {𝐴})
24	22, 23	jaoi 853	. . . 4 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
25	19, 24	impbii 208	. . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴}))
26	25	abbii 2800	. 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
27		df-pw 4603	. 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}}
28		dfpr2 4646	. 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
29	26, 27, 28	3eqtr4i 2768	1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∨ wo 843  ∀wal 1537   = wceq 1539  ∃wex 1779   ∈ wcel 2104  {cab 2707   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  {csn 4627  {cpr 4629

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-sn 4628  df-pr 4630

This theorem is referenced by: (None)