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Theorem altxpsspw 35801
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 35799 . . 3 (𝑧 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫)
2 df-altop 35782 . . . . . 6 𝑥, 𝑦⟫ = {{𝑥}, {𝑥, {𝑦}}}
3 snssi 4817 . . . . . . . . 9 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 ssun3 4175 . . . . . . . . 9 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
53, 4syl 17 . . . . . . . 8 (𝑥𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
65adantr 479 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
7 elun1 4177 . . . . . . . . 9 (𝑥𝐴𝑥 ∈ (𝐴 ∪ 𝒫 𝐵))
8 snssi 4817 . . . . . . . . . 10 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
9 vsnex 5435 . . . . . . . . . . . 12 {𝑦} ∈ V
109elpw 4611 . . . . . . . . . . 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 611 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → (𝑥 ∈ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵)))
15 vex 3466 . . . . . . . . 9 𝑥 ∈ V
1615, 9prss 4829 . . . . . . . 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 5438 . . . . . . . . 9 {{𝑥}, {𝑥, {𝑦}}} ∈ V
1918elpw 4611 . . . . . . . 8 ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ {{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵))
20 vsnex 5435 . . . . . . . . 9 {𝑥} ∈ V
21 prex 5438 . . . . . . . . 9 {𝑥, {𝑦}} ∈ V
2220, 21prsspw 4852 . . . . . . . 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 581 . . . . . 6 ((𝑥𝐴𝑦𝐵) → {{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
252, 24eqeltrid 2830 . . . . 5 ((𝑥𝐴𝑦𝐵) → ⟪𝑥, 𝑦⟫ ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
26 eleq1a 2821 . . . . 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 3190 . . 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 3983 1 (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wrex 3060  cun 3945  wss 3947  𝒫 cpw 4607  {csn 4633  {cpr 4635  caltop 35780   ×× caltxp 35781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rex 3061  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-pw 4609  df-sn 4634  df-pr 4636  df-altop 35782  df-altxp 35783
This theorem is referenced by:  altxpexg  35802
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