Theorem altxpsspw 31254

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 31252	. . 3 ⊢ (𝑧 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫)
2		df-altop 31235	. . . . . 6 ⊢ ⟪𝑥, 𝑦⟫ = {{𝑥}, {𝑥, {𝑦}}}
3		snssi 4280	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴)
4		ssun3 3740	. . . . . . . . 9 ⊢ ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
6	5	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
7		elun1 3742	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 ∪ 𝒫 𝐵))
8		snssi 4280	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ⊆ 𝐵)
9		snex 4835	. . . . . . . . . . . 12 ⊢ {𝑦} ∈ V
10	9	elpw 4114	. . . . . . . . . . 11 ⊢ ({𝑦} ∈ 𝒫 𝐵 ↔ {𝑦} ⊆ 𝐵)
11		elun2 3743	. . . . . . . . . . 11 ⊢ ({𝑦} ∈ 𝒫 𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
12	10, 11	sylbir 224	. . . . . . . . . 10 ⊢ ({𝑦} ⊆ 𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
13	8, 12	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
14	7, 13	anim12i 588	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵)))
15		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
16	15, 9	prss 4291	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵)) ↔ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵))
17	14, 16	sylib 207	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵))
18		prex 4836	. . . . . . . . 9 ⊢ {{𝑥}, {𝑥, {𝑦}}} ∈ V
19	18	elpw 4114	. . . . . . . 8 ⊢ ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ {{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵))
20		snex 4835	. . . . . . . . 9 ⊢ {𝑥} ∈ V
21		prex 4836	. . . . . . . . 9 ⊢ {𝑥, {𝑦}} ∈ V
22	20, 21	prsspw 4316	. . . . . . . 8 ⊢ ({{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵)))
23	19, 22	bitri 263	. . . . . . 7 ⊢ ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵)))
24	6, 17, 23	sylanbrc 695	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
25	2, 24	syl5eqel 2692	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟪𝑥, 𝑦⟫ ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
26		eleq1a 2683	. . . . 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 3018	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫ → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
29	1, 28	sylbi 206	. 2 ⊢ (𝑧 ∈ (𝐴 ×× 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
30	29	ssriv 3572	1 ⊢ (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  {cpr 4127  ⟪caltop 31233   ×× caltxp 31234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-altop 31235  df-altxp 31236

This theorem is referenced by:  altxpexg  31255