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Theorem xpsspw 5156
 Description: A Cartesian product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.)
Assertion
Ref Expression
xpsspw (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)

Proof of Theorem xpsspw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5150 . 2 Rel (𝐴 × 𝐵)
2 opelxp 5070 . . 3 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 snssi 4280 . . . . . . . 8 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
4 ssun3 3740 . . . . . . . 8 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴𝐵))
53, 4syl 17 . . . . . . 7 (𝑥𝐴 → {𝑥} ⊆ (𝐴𝐵))
6 snex 4835 . . . . . . . 8 {𝑥} ∈ V
76elpw 4114 . . . . . . 7 ({𝑥} ∈ 𝒫 (𝐴𝐵) ↔ {𝑥} ⊆ (𝐴𝐵))
85, 7sylibr 223 . . . . . 6 (𝑥𝐴 → {𝑥} ∈ 𝒫 (𝐴𝐵))
98adantr 480 . . . . 5 ((𝑥𝐴𝑦𝐵) → {𝑥} ∈ 𝒫 (𝐴𝐵))
10 df-pr 4128 . . . . . . 7 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
11 snssi 4280 . . . . . . . . . 10 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
12 ssun4 3741 . . . . . . . . . 10 ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴𝐵))
1311, 12syl 17 . . . . . . . . 9 (𝑦𝐵 → {𝑦} ⊆ (𝐴𝐵))
145, 13anim12i 588 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → ({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)))
15 unss 3749 . . . . . . . 8 (({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1614, 15sylib 207 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1710, 16syl5eqss 3612 . . . . . 6 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ⊆ (𝐴𝐵))
18 zfpair2 4834 . . . . . . 7 {𝑥, 𝑦} ∈ V
1918elpw 4114 . . . . . 6 ({𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴𝐵))
2017, 19sylibr 223 . . . . 5 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵))
219, 20jca 553 . . . 4 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)))
22 prex 4836 . . . . . 6 {{𝑥}, {𝑥, 𝑦}} ∈ V
2322elpw 4114 . . . . 5 ({{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
24 vex 3176 . . . . . . 7 𝑥 ∈ V
25 vex 3176 . . . . . . 7 𝑦 ∈ V
2624, 25dfop 4339 . . . . . 6 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
2726eleq1i 2679 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵) ↔ {{𝑥}, {𝑥, 𝑦}} ∈ 𝒫 𝒫 (𝐴𝐵))
286, 18prss 4291 . . . . 5 (({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)) ↔ {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
2923, 27, 283bitr4ri 292 . . . 4 (({𝑥} ∈ 𝒫 (𝐴𝐵) ∧ {𝑥, 𝑦} ∈ 𝒫 (𝐴𝐵)) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
3021, 29sylib 207 . . 3 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
312, 30sylbi 206 . 2 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ 𝒫 𝒫 (𝐴𝐵))
321, 31relssi 5134 1 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  {cpr 4127  ⟨cop 4131   × cxp 5036 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-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045 This theorem is referenced by:  unixpss  5157  xpexg  6858  rankxpu  8622  wunxp  9425  gruxp  9508
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