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Theorem altxpsspw 35018
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.
StepHypRef Expression
1 elaltxp 35016 . . 3 (𝑧 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫)
2 df-altop 34999 . . . . . 6 𝑥, 𝑦⟫ = {{𝑥}, {𝑥, {𝑦}}}
3 snssi 4811 . . . . . . . . 9 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 ssun3 4174 . . . . . . . . 9 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
53, 4syl 17 . . . . . . . 8 (𝑥𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
65adantr 481 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
7 elun1 4176 . . . . . . . . 9 (𝑥𝐴𝑥 ∈ (𝐴 ∪ 𝒫 𝐵))
8 snssi 4811 . . . . . . . . . 10 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
9 vsnex 5429 . . . . . . . . . . . 12 {𝑦} ∈ V
109elpw 4606 . . . . . . . . . . 11 ({𝑦} ∈ 𝒫 𝐵 ↔ {𝑦} ⊆ 𝐵)
11 elun2 4177 . . . . . . . . . . 11 ({𝑦} ∈ 𝒫 𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
1210, 11sylbir 234 . . . . . . . . . 10 ({𝑦} ⊆ 𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
138, 12syl 17 . . . . . . . . 9 (𝑦𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
147, 13anim12i 613 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → (𝑥 ∈ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵)))
15 vex 3478 . . . . . . . . 9 𝑥 ∈ V
1615, 9prss 4823 . . . . . . . 8 ((𝑥 ∈ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵)) ↔ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵))
1714, 16sylib 217 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵))
18 prex 5432 . . . . . . . . 9 {{𝑥}, {𝑥, {𝑦}}} ∈ V
1918elpw 4606 . . . . . . . 8 ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ {{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵))
20 vsnex 5429 . . . . . . . . 9 {𝑥} ∈ V
21 prex 5432 . . . . . . . . 9 {𝑥, {𝑦}} ∈ V
2220, 21prsspw 4846 . . . . . . . 8 ({{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵)))
2319, 22bitri 274 . . . . . . 7 ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵)))
246, 17, 23sylanbrc 583 . . . . . 6 ((𝑥𝐴𝑦𝐵) → {{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
252, 24eqeltrid 2837 . . . . 5 ((𝑥𝐴𝑦𝐵) → ⟪𝑥, 𝑦⟫ ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
26 eleq1a 2828 . . . . 5 (⟪𝑥, 𝑦⟫ ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) → (𝑧 = ⟪𝑥, 𝑦⟫ → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)))
2725, 26syl 17 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝑧 = ⟪𝑥, 𝑦⟫ → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)))
2827rexlimivv 3199 . . 3 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫ → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
291, 28sylbi 216 . 2 (𝑧 ∈ (𝐴 ×× 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
3029ssriv 3986 1 (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3070  cun 3946  wss 3948  𝒫 cpw 4602  {csn 4628  {cpr 4630  caltop 34997   ×× caltxp 34998
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-pw 4604  df-sn 4629  df-pr 4631  df-altop 34999  df-altxp 35000
This theorem is referenced by:  altxpexg  35019
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