Theorem xpsspw 5758

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 5642	. 2 ⊢ Rel (𝐴 × 𝐵)
2		opelxp 5660	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3		snssi 4752	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
4		ssun3 4121	. . . . . . . 8 ⊢ ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝐵))
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝐵))
6		vsnex 5372	. . . . . . . 8 ⊢ {𝑥} ∈ V
7	6	elpw 4546	. . . . . . 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 4571	. . . . . . 7 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
11		snssi 4752	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ 𝐵)
12		ssun4 4122	. . . . . . . . . 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 4131	. . . . . . . 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 3961	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ⊆ (𝐴 ∪ 𝐵))
18		zfpair2 5371	. . . . . . 7 ⊢ {𝑥, 𝑦} ∈ V
19	18	elpw 4546	. . . . . 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 5375	. . . . . 6 ⊢ {{𝑥}, {𝑥, 𝑦}} ∈ V
23	22	elpw 4546	. . . . 5 ⊢ ({{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴 ∪ 𝐵))
24		vex 3434	. . . . . . 7 ⊢ 𝑥 ∈ V
25		vex 3434	. . . . . . 7 ⊢ 𝑦 ∈ V
26	24, 25	dfop 4816	. . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
27	26	eleq1i 2828	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝐵))
28	6, 18	prss 4764	. . . . 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 5736	1 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)

Syntax hints:   ∧ wa 395   ∈ wcel 2114   ∪ cun 3888   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568  {cpr 4570  ⟨cop 4574   × cxp 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-opab 5149  df-xp 5630  df-rel 5631

This theorem is referenced by:  unixpss  5759  xpexg  7697  xpfi  9223  rankxpu  9791  wunxp  10638  gruxp  10721  xpwf  45409