Theorem xpsspw 5807

Description: A Cartesian product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)

Assertion

Ref	Expression
xpsspw	⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)

Proof of Theorem xpsspw

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

Step	Hyp	Ref	Expression
1		relxp 5693	. 2 ⊢ Rel (𝐴 × 𝐵)
2		opelxp 5711	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		snssi 4810	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
4		ssun3 4173	. . . . . . . 8 ⊢ ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝐵))
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝐵))
6		vsnex 5428	. . . . . . . 8 ⊢ {𝑥} ∈ V
7	6	elpw 4605	. . . . . . 7 ⊢ ({𝑥} ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ {𝑥} ⊆ (𝐴 ∪ 𝐵))
8	5, 7	sylibr 233	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 (𝐴 ∪ 𝐵))
9	8	adantr 482	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥} ∈ 𝒫 (𝐴 ∪ 𝐵))
10		df-pr 4630	. . . . . . 7 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
11		snssi 4810	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ 𝐵)
12		ssun4 4174	. . . . . . . . . 10 ⊢ ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴 ∪ 𝐵))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ (𝐴 ∪ 𝐵))
14	5, 13	anim12i 614	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ⊆ (𝐴 ∪ 𝐵) ∧ {𝑦} ⊆ (𝐴 ∪ 𝐵)))
15		unss 4183	. . . . . . . 8 ⊢ (({𝑥} ⊆ (𝐴 ∪ 𝐵) ∧ {𝑦} ⊆ (𝐴 ∪ 𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
16	14, 15	sylib 217	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
17	10, 16	eqsstrid 4029	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵))
18		zfpair2 5427	. . . . . . 7 ⊢ {𝑥, 𝑦} ∈ V
19	18	elpw 4605	. . . . . 6 ⊢ ({𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵))
20	17, 19	sylibr 233	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵))
21	9, 20	jca 513	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ∈ 𝒫 (𝐴 ∪ 𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵)))
22		prex 5431	. . . . . 6 ⊢ {{𝑥}, {𝑥, 𝑦}} ∈ V
23	22	elpw 4605	. . . . 5 ⊢ ({{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴 ∪ 𝐵))
24		vex 3479	. . . . . . 7 ⊢ 𝑥 ∈ V
25		vex 3479	. . . . . . 7 ⊢ 𝑦 ∈ V
26	24, 25	dfop 4871	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
27	26	eleq1i 2825	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵))
28	6, 18	prss 4822	. . . . 5 ⊢ (({𝑥} ∈ 𝒫 (𝐴 ∪ 𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵)) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴 ∪ 𝐵))
29	23, 27, 28	3bitr4ri 304	. . . 4 ⊢ (({𝑥} ∈ 𝒫 (𝐴 ∪ 𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵))
30	21, 29	sylib 217	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵))
31	2, 30	sylbi 216	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵))
32	1, 31	relssi 5785	1 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 397   ∈ wcel 2107   ∪ cun 3945   ⊆ wss 3947  𝒫 cpw 4601  {csn 4627  {cpr 4629  ⟨cop 4633   × cxp 5673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-opab 5210  df-xp 5681  df-rel 5682

This theorem is referenced by:  unixpss  5808  xpexg  7732  xpfi  9313  rankxpu  9867  wunxp  10715  gruxp  10798