Theorem altxpsspw 34370

Description: An inclusion rule for alternate Cartesian products. (Contributed by Scott Fenton, 24-Mar-2012.)

Assertion

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

Proof of Theorem altxpsspw

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

Step	Hyp	Ref	Expression
1		elaltxp 34368	. . 3 ⊢ (𝑧 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫)
2		df-altop 34351	. . . . . 6 ⊢ ⟪𝑥, 𝑦⟫ = {{𝑥}, {𝑥, {𝑦}}}
3		snssi 4754	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
4		ssun3 4120	. . . . . . . . 9 ⊢ ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
6	5	adantr 481	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
7		elun1 4122	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∪ 𝒫 𝐵))
8		snssi 4754	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ 𝐵)
9		vsnex 5371	. . . . . . . . . . . 12 ⊢ {𝑦} ∈ V
10	9	elpw 4550	. . . . . . . . . . 11 ⊢ ({𝑦} ∈ 𝒫 𝐵 ↔ {𝑦} ⊆ 𝐵)
11		elun2 4123	. . . . . . . . . . 11 ⊢ ({𝑦} ∈ 𝒫 𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
12	10, 11	sylbir 234	. . . . . . . . . 10 ⊢ ({𝑦} ⊆ 𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
13	8, 12	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
14	7, 13	anim12i 613	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵)))
15		vex 3445	. . . . . . . . 9 ⊢ 𝑥 ∈ V
16	15, 9	prss 4766	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵)) ↔ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵))
17	14, 16	sylib 217	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵))
18		prex 5374	. . . . . . . . 9 ⊢ {{𝑥}, {𝑥, {𝑦}}} ∈ V
19	18	elpw 4550	. . . . . . . 8 ⊢ ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ {{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵))
20		vsnex 5371	. . . . . . . . 9 ⊢ {𝑥} ∈ V
21		prex 5374	. . . . . . . . 9 ⊢ {𝑥, {𝑦}} ∈ V
22	20, 21	prsspw 4789	. . . . . . . 8 ⊢ ({{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵)))
23	19, 22	bitri 274	. . . . . . 7 ⊢ ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵)))
24	6, 17, 23	sylanbrc 583	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
25	2, 24	eqeltrid 2841	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟪𝑥, 𝑦⟫ ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
26		eleq1a 2832	. . . . 5 ⊢ (⟪𝑥, 𝑦⟫ ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) → (𝑧 = ⟪𝑥, 𝑦⟫ → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)))
27	25, 26	syl 17	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑧 = ⟪𝑥, 𝑦⟫ → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)))
28	27	rexlimivv 3192	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫ → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
29	1, 28	sylbi 216	. 2 ⊢ (𝑧 ∈ (𝐴 ×× 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
30	29	ssriv 3935	1 ⊢ (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∃wrex 3070   ∪ cun 3895   ⊆ wss 3897  𝒫 cpw 4546  {csn 4572  {cpr 4574  ⟪caltop 34349   ×× caltxp 34350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rex 3071  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-pw 4548  df-sn 4573  df-pr 4575  df-altop 34351  df-altxp 34352

This theorem is referenced by:  altxpexg  34371