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Theorem xpsspw 4721
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 4625 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2 vex 2733 . . . . . . . 8 𝑥 ∈ V
3 vex 2733 . . . . . . . 8 𝑦 ∈ V
42, 3dfop 3762 . . . . . . 7 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
5 snssi 3722 . . . . . . . . . . . . 13 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
6 ssun3 3292 . . . . . . . . . . . . 13 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴𝐵))
75, 6syl 14 . . . . . . . . . . . 12 (𝑥𝐴 → {𝑥} ⊆ (𝐴𝐵))
87adantr 274 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → {𝑥} ⊆ (𝐴𝐵))
9 sseq1 3170 . . . . . . . . . . 11 (𝑧 = {𝑥} → (𝑧 ⊆ (𝐴𝐵) ↔ {𝑥} ⊆ (𝐴𝐵)))
108, 9syl5ibrcom 156 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑧 = {𝑥} → 𝑧 ⊆ (𝐴𝐵)))
11 df-pr 3588 . . . . . . . . . . . 12 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
12 snssi 3722 . . . . . . . . . . . . . . 15 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
13 ssun4 3293 . . . . . . . . . . . . . . 15 ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴𝐵))
1412, 13syl 14 . . . . . . . . . . . . . 14 (𝑦𝐵 → {𝑦} ⊆ (𝐴𝐵))
157, 14anim12i 336 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐵) → ({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)))
16 unss 3301 . . . . . . . . . . . . 13 (({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1715, 16sylib 121 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1811, 17eqsstrid 3193 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ⊆ (𝐴𝐵))
19 sseq1 3170 . . . . . . . . . . 11 (𝑧 = {𝑥, 𝑦} → (𝑧 ⊆ (𝐴𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴𝐵)))
2018, 19syl5ibrcom 156 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑧 = {𝑥, 𝑦} → 𝑧 ⊆ (𝐴𝐵)))
2110, 20jaod 712 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ((𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}) → 𝑧 ⊆ (𝐴𝐵)))
22 vex 2733 . . . . . . . . . 10 𝑧 ∈ V
2322elpr 3602 . . . . . . . . 9 (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
2422elpw 3570 . . . . . . . . 9 (𝑧 ∈ 𝒫 (𝐴𝐵) ↔ 𝑧 ⊆ (𝐴𝐵))
2521, 23, 243imtr4g 204 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} → 𝑧 ∈ 𝒫 (𝐴𝐵)))
2625ssrdv 3153 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
274, 26eqsstrid 3193 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵))
28 sseq1 3170 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ⊆ 𝒫 (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵)))
2928biimpar 295 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3027, 29sylan2 284 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3130exlimivv 1889 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
321, 31syl 14 . . 3 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3322elpw 3570 . . 3 (𝑧 ∈ 𝒫 𝒫 (𝐴𝐵) ↔ 𝑧 ⊆ 𝒫 (𝐴𝐵))
3432, 33sylibr 133 . 2 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴𝐵))
3534ssriv 3151 1 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103  wo 703   = wceq 1348  wex 1485  wcel 2141  cun 3119  wss 3121  𝒫 cpw 3564  {csn 3581  {cpr 3582  cop 3584   × cxp 4607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4049  df-xp 4615
This theorem is referenced by:  unixpss  4722  xpexg  4723
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