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Theorem altxpsspw 34949
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 34947 . . 3 (𝑧 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫)
2 df-altop 34930 . . . . . 6 𝑥, 𝑦⟫ = {{𝑥}, {𝑥, {𝑦}}}
3 snssi 4812 . . . . . . . . 9 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 ssun3 4175 . . . . . . . . 9 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
53, 4syl 17 . . . . . . . 8 (𝑥𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
65adantr 482 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
7 elun1 4177 . . . . . . . . 9 (𝑥𝐴𝑥 ∈ (𝐴 ∪ 𝒫 𝐵))
8 snssi 4812 . . . . . . . . . 10 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
9 vsnex 5430 . . . . . . . . . . . 12 {𝑦} ∈ V
109elpw 4607 . . . . . . . . . . 11 ({𝑦} ∈ 𝒫 𝐵 ↔ {𝑦} ⊆ 𝐵)
11 elun2 4178 . . . . . . . . . . 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 614 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → (𝑥 ∈ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵)))
15 vex 3479 . . . . . . . . 9 𝑥 ∈ V
1615, 9prss 4824 . . . . . . . 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 5433 . . . . . . . . 9 {{𝑥}, {𝑥, {𝑦}}} ∈ V
1918elpw 4607 . . . . . . . 8 ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ {{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵))
20 vsnex 5430 . . . . . . . . 9 {𝑥} ∈ V
21 prex 5433 . . . . . . . . 9 {𝑥, {𝑦}} ∈ V
2220, 21prsspw 4847 . . . . . . . 8 ({{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵)))
2319, 22bitri 275 . . . . . . 7 ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵)))
246, 17, 23sylanbrc 584 . . . . . 6 ((𝑥𝐴𝑦𝐵) → {{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
252, 24eqeltrid 2838 . . . . 5 ((𝑥𝐴𝑦𝐵) → ⟪𝑥, 𝑦⟫ ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
26 eleq1a 2829 . . . . 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 3200 . . 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 3987 1 (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wrex 3071  cun 3947  wss 3949  𝒫 cpw 4603  {csn 4629  {cpr 4631  caltop 34928   ×× caltxp 34929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-pw 4605  df-sn 4630  df-pr 4632  df-altop 34930  df-altxp 34931
This theorem is referenced by:  altxpexg  34950
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