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Theorem altxpsspw 32421
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 32419 . . 3 (𝑧 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫)
2 df-altop 32402 . . . . . 6 𝑥, 𝑦⟫ = {{𝑥}, {𝑥, {𝑦}}}
3 snssi 4474 . . . . . . . . 9 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 ssun3 3929 . . . . . . . . 9 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
53, 4syl 17 . . . . . . . 8 (𝑥𝐴 → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
65adantr 466 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵))
7 elun1 3931 . . . . . . . . 9 (𝑥𝐴𝑥 ∈ (𝐴 ∪ 𝒫 𝐵))
8 snssi 4474 . . . . . . . . . 10 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
9 snex 5036 . . . . . . . . . . . 12 {𝑦} ∈ V
109elpw 4303 . . . . . . . . . . 11 ({𝑦} ∈ 𝒫 𝐵 ↔ {𝑦} ⊆ 𝐵)
11 elun2 3932 . . . . . . . . . . 11 ({𝑦} ∈ 𝒫 𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
1210, 11sylbir 225 . . . . . . . . . 10 ({𝑦} ⊆ 𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
138, 12syl 17 . . . . . . . . 9 (𝑦𝐵 → {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵))
147, 13anim12i 600 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → (𝑥 ∈ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵)))
15 vex 3354 . . . . . . . . 9 𝑥 ∈ V
1615, 9prss 4486 . . . . . . . 8 ((𝑥 ∈ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑦} ∈ (𝐴 ∪ 𝒫 𝐵)) ↔ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵))
1714, 16sylib 208 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵))
18 prex 5037 . . . . . . . . 9 {{𝑥}, {𝑥, {𝑦}}} ∈ V
1918elpw 4303 . . . . . . . 8 ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ {{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵))
20 snex 5036 . . . . . . . . 9 {𝑥} ∈ V
21 prex 5037 . . . . . . . . 9 {𝑥, {𝑦}} ∈ V
2220, 21prsspw 4508 . . . . . . . 8 ({{𝑥}, {𝑥, {𝑦}}} ⊆ 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵)))
2319, 22bitri 264 . . . . . . 7 ({{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ↔ ({𝑥} ⊆ (𝐴 ∪ 𝒫 𝐵) ∧ {𝑥, {𝑦}} ⊆ (𝐴 ∪ 𝒫 𝐵)))
246, 17, 23sylanbrc 572 . . . . . 6 ((𝑥𝐴𝑦𝐵) → {{𝑥}, {𝑥, {𝑦}}} ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
252, 24syl5eqel 2854 . . . . 5 ((𝑥𝐴𝑦𝐵) → ⟪𝑥, 𝑦⟫ ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
26 eleq1a 2845 . . . . 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 3184 . . 3 (∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫ → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
291, 28sylbi 207 . 2 (𝑧 ∈ (𝐴 ×× 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵))
3029ssriv 3756 1 (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wrex 3062  cun 3721  wss 3723  𝒫 cpw 4297  {csn 4316  {cpr 4318  caltop 32400   ×× caltxp 32401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-pr 4319  df-altop 32402  df-altxp 32403
This theorem is referenced by:  altxpexg  32422
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