Theorem pwssun 5550

Description: The power class of the union of two classes is a subset of the union of their power classes, iff one class is a subclass of the other. Exercise 4.12(l) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.)

Assertion

Ref	Expression
pwssun	⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))

Proof of Theorem pwssun

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

Step	Hyp	Ref	Expression
1		ssequn2 4169	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
2		pweq 4594	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → 𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐴)
3		eqimss 4022	. . . . . . 7 ⊢ (𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐴 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐴 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴)
5	1, 4	sylbi 217	. . . . 5 ⊢ (𝐵 ⊆ 𝐴 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴)
6		ssequn1 4166	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
7		pweq 4594	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → 𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐵)
8		eqimss 4022	. . . . . . 7 ⊢ (𝒫 (𝐴 ∪ 𝐵) = 𝒫 𝐵 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵)
9	7, 8	syl 17	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵)
10	6, 9	sylbi 217	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵)
11	5, 10	orim12i 908	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∨ 𝐴 ⊆ 𝐵) → (𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵))
12	11	orcoms 872	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → (𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵))
13		ssun 4175	. . 3 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐴 ∨ 𝒫 (𝐴 ∪ 𝐵) ⊆ 𝒫 𝐵) → 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))
14	12, 13	syl 17	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) → 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))
15		vex 3468	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
16	15	snss 4766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
17		vex 3468	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑦 ∈ V
18	17	snss 4766	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐵 ↔ {𝑦} ⊆ 𝐵)
19		unss12 4168	. . . . . . . . . . . . . . . . . . 19 ⊢ (({𝑥} ⊆ 𝐴 ∧ {𝑦} ⊆ 𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
20	16, 18, 19	syl2anb 598	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
21		zfpair2 5408	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑥, 𝑦} ∈ V
22	21	elpw 4584	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵))
23		df-pr 4609	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
24	23	sseq1i 3992	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
25	22, 24	bitr2i 276	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵) ↔ {𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵))
26	20, 25	sylib 218	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵))
27		ssel 3957	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) → ({𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵) → {𝑥, 𝑦} ∈ (𝒫 𝐴 ∪ 𝒫 𝐵)))
28	26, 27	syl5 34	. . . . . . . . . . . . . . . 16 ⊢ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ (𝒫 𝐴 ∪ 𝒫 𝐵)))
29	28	expcomd 416	. . . . . . . . . . . . . . 15 ⊢ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → {𝑥, 𝑦} ∈ (𝒫 𝐴 ∪ 𝒫 𝐵))))
30	29	imp31 417	. . . . . . . . . . . . . 14 ⊢ (((𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑦} ∈ (𝒫 𝐴 ∪ 𝒫 𝐵))
31		elun 4133	. . . . . . . . . . . . . 14 ⊢ ({𝑥, 𝑦} ∈ (𝒫 𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥, 𝑦} ∈ 𝒫 𝐴 ∨ {𝑥, 𝑦} ∈ 𝒫 𝐵))
32	30, 31	sylib 218	. . . . . . . . . . . . 13 ⊢ (((𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ({𝑥, 𝑦} ∈ 𝒫 𝐴 ∨ {𝑥, 𝑦} ∈ 𝒫 𝐵))
33	21	elpw 4584	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥, 𝑦} ∈ 𝒫 𝐴 ↔ {𝑥, 𝑦} ⊆ 𝐴)
34	15, 17	prss 4801	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
35	33, 34	bitr4i 278	. . . . . . . . . . . . . . 15 ⊢ ({𝑥, 𝑦} ∈ 𝒫 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
36	35	simprbi 496	. . . . . . . . . . . . . 14 ⊢ ({𝑥, 𝑦} ∈ 𝒫 𝐴 → 𝑦 ∈ 𝐴)
37	21	elpw 4584	. . . . . . . . . . . . . . . 16 ⊢ ({𝑥, 𝑦} ∈ 𝒫 𝐵 ↔ {𝑥, 𝑦} ⊆ 𝐵)
38	15, 17	prss 4801	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
39	37, 38	bitr4i 278	. . . . . . . . . . . . . . 15 ⊢ ({𝑥, 𝑦} ∈ 𝒫 𝐵 ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
40	39	simplbi 497	. . . . . . . . . . . . . 14 ⊢ ({𝑥, 𝑦} ∈ 𝒫 𝐵 → 𝑥 ∈ 𝐵)
41	36, 40	orim12i 908	. . . . . . . . . . . . 13 ⊢ (({𝑥, 𝑦} ∈ 𝒫 𝐴 ∨ {𝑥, 𝑦} ∈ 𝒫 𝐵) → (𝑦 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
42	32, 41	syl 17	. . . . . . . . . . . 12 ⊢ (((𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
43	42	ord 864	. . . . . . . . . . 11 ⊢ (((𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑦 ∈ 𝐴 → 𝑥 ∈ 𝐵))
44	43	impancom 451	. . . . . . . . . 10 ⊢ (((𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
45	44	ssrdv 3969	. . . . . . . . 9 ⊢ (((𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ¬ 𝑦 ∈ 𝐴) → 𝐴 ⊆ 𝐵)
46	45	exp31 419	. . . . . . . 8 ⊢ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) → (𝑦 ∈ 𝐵 → (¬ 𝑦 ∈ 𝐴 → 𝐴 ⊆ 𝐵)))
47		con1b 358	. . . . . . . 8 ⊢ ((¬ 𝑦 ∈ 𝐴 → 𝐴 ⊆ 𝐵) ↔ (¬ 𝐴 ⊆ 𝐵 → 𝑦 ∈ 𝐴))
48	46, 47	imbitrdi 251	. . . . . . 7 ⊢ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) → (𝑦 ∈ 𝐵 → (¬ 𝐴 ⊆ 𝐵 → 𝑦 ∈ 𝐴)))
49	48	com23 86	. . . . . 6 ⊢ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) → (¬ 𝐴 ⊆ 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)))
50	49	imp 406	. . . . 5 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ ¬ 𝐴 ⊆ 𝐵) → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴))
51	50	ssrdv 3969	. . . 4 ⊢ ((𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) ∧ ¬ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ 𝐴)
52	51	ex 412	. . 3 ⊢ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) → (¬ 𝐴 ⊆ 𝐵 → 𝐵 ⊆ 𝐴))
53	52	orrd 863	. 2 ⊢ (𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
54	14, 53	impbii 209	1 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ 𝒫 (𝐴 ∪ 𝐵) ⊆ (𝒫 𝐴 ∪ 𝒫 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ∪ cun 3929   ⊆ wss 3931  𝒫 cpw 4580  {csn 4606  {cpr 4608

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-ss 3948  df-pw 4582  df-sn 4607  df-pr 4609

This theorem is referenced by:  pwun  5551