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Theorem xpsspw 4538
Description: A cross 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 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxpi 4444 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2 vex 2622 . . . . . . . 8 𝑥 ∈ V
3 vex 2622 . . . . . . . 8 𝑦 ∈ V
42, 3dfop 3616 . . . . . . 7 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
5 snssi 3576 . . . . . . . . . . . . 13 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
6 ssun3 3163 . . . . . . . . . . . . 13 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴𝐵))
75, 6syl 14 . . . . . . . . . . . 12 (𝑥𝐴 → {𝑥} ⊆ (𝐴𝐵))
87adantr 270 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → {𝑥} ⊆ (𝐴𝐵))
9 sseq1 3045 . . . . . . . . . . 11 (𝑧 = {𝑥} → (𝑧 ⊆ (𝐴𝐵) ↔ {𝑥} ⊆ (𝐴𝐵)))
108, 9syl5ibrcom 155 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑧 = {𝑥} → 𝑧 ⊆ (𝐴𝐵)))
11 df-pr 3448 . . . . . . . . . . . 12 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
12 snssi 3576 . . . . . . . . . . . . . . 15 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
13 ssun4 3164 . . . . . . . . . . . . . . 15 ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴𝐵))
1412, 13syl 14 . . . . . . . . . . . . . 14 (𝑦𝐵 → {𝑦} ⊆ (𝐴𝐵))
157, 14anim12i 331 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐵) → ({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)))
16 unss 3172 . . . . . . . . . . . . 13 (({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1715, 16sylib 120 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1811, 17syl5eqss 3068 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ⊆ (𝐴𝐵))
19 sseq1 3045 . . . . . . . . . . 11 (𝑧 = {𝑥, 𝑦} → (𝑧 ⊆ (𝐴𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴𝐵)))
2018, 19syl5ibrcom 155 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑧 = {𝑥, 𝑦} → 𝑧 ⊆ (𝐴𝐵)))
2110, 20jaod 672 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ((𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}) → 𝑧 ⊆ (𝐴𝐵)))
22 vex 2622 . . . . . . . . . 10 𝑧 ∈ V
2322elpr 3462 . . . . . . . . 9 (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
2422elpw 3431 . . . . . . . . 9 (𝑧 ∈ 𝒫 (𝐴𝐵) ↔ 𝑧 ⊆ (𝐴𝐵))
2521, 23, 243imtr4g 203 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} → 𝑧 ∈ 𝒫 (𝐴𝐵)))
2625ssrdv 3029 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
274, 26syl5eqss 3068 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵))
28 sseq1 3045 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ⊆ 𝒫 (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵)))
2928biimpar 291 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3027, 29sylan2 280 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3130exlimivv 1824 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
321, 31syl 14 . . 3 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3322elpw 3431 . . 3 (𝑧 ∈ 𝒫 𝒫 (𝐴𝐵) ↔ 𝑧 ⊆ 𝒫 (𝐴𝐵))
3432, 33sylibr 132 . 2 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴𝐵))
3534ssriv 3027 1 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 102  wo 664   = wceq 1289  wex 1426  wcel 1438  cun 2995  wss 2997  𝒫 cpw 3425  {csn 3441  {cpr 3442  cop 3444   × cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434
This theorem is referenced by:  unixpss  4539  xpexg  4540
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