Theorem xpsspw 5775

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 5659	. 2 ⊢ Rel (𝐴 × 𝐵)
2		opelxp 5677	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		snssi 4775	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
4		ssun3 4146	. . . . . . . 8 ⊢ ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝐵))
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝐵))
6		vsnex 5392	. . . . . . . 8 ⊢ {𝑥} ∈ V
7	6	elpw 4570	. . . . . . 7 ⊢ ({𝑥} ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ {𝑥} ⊆ (𝐴 ∪ 𝐵))
8	5, 7	sylibr 234	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ∈ 𝒫 (𝐴 ∪ 𝐵))
9	8	adantr 480	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥} ∈ 𝒫 (𝐴 ∪ 𝐵))
10		df-pr 4595	. . . . . . 7 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
11		snssi 4775	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ 𝐵)
12		ssun4 4147	. . . . . . . . . 10 ⊢ ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴 ∪ 𝐵))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ (𝐴 ∪ 𝐵))
14	5, 13	anim12i 613	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ⊆ (𝐴 ∪ 𝐵) ∧ {𝑦} ⊆ (𝐴 ∪ 𝐵)))
15		unss 4156	. . . . . . . 8 ⊢ (({𝑥} ⊆ (𝐴 ∪ 𝐵) ∧ {𝑦} ⊆ (𝐴 ∪ 𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
16	14, 15	sylib 218	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴 ∪ 𝐵))
17	10, 16	eqsstrid 3988	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵))
18		zfpair2 5391	. . . . . . 7 ⊢ {𝑥, 𝑦} ∈ V
19	18	elpw 4570	. . . . . 6 ⊢ ({𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵))
20	17, 19	sylibr 234	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵))
21	9, 20	jca 511	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ({𝑥} ∈ 𝒫 (𝐴 ∪ 𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴 ∪ 𝐵)))
22		prex 5395	. . . . . 6 ⊢ {{𝑥}, {𝑥, 𝑦}} ∈ V
23	22	elpw 4570	. . . . 5 ⊢ ({{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴 ∪ 𝐵))
24		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
25		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
26	24, 25	dfop 4839	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
27	26	eleq1i 2820	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵))
28	6, 18	prss 4787	. . . . 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 218	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵))
31	2, 30	sylbi 217	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵))
32	1, 31	relssi 5753	1 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 395   ∈ wcel 2109   ∪ cun 3915   ⊆ wss 3917  𝒫 cpw 4566  {csn 4592  {cpr 4594  ⟨cop 4598   × cxp 5639

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-opab 5173  df-xp 5647  df-rel 5648

This theorem is referenced by:  unixpss  5776  xpexg  7729  xpfi  9276  rankxpu  9836  wunxp  10684  gruxp  10767  xpwf  44961