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Theorem xpsspw 4651
 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 4555 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2 vex 2689 . . . . . . . 8 𝑥 ∈ V
3 vex 2689 . . . . . . . 8 𝑦 ∈ V
42, 3dfop 3704 . . . . . . 7 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
5 snssi 3664 . . . . . . . . . . . . 13 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
6 ssun3 3241 . . . . . . . . . . . . 13 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴𝐵))
75, 6syl 14 . . . . . . . . . . . 12 (𝑥𝐴 → {𝑥} ⊆ (𝐴𝐵))
87adantr 274 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → {𝑥} ⊆ (𝐴𝐵))
9 sseq1 3120 . . . . . . . . . . 11 (𝑧 = {𝑥} → (𝑧 ⊆ (𝐴𝐵) ↔ {𝑥} ⊆ (𝐴𝐵)))
108, 9syl5ibrcom 156 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑧 = {𝑥} → 𝑧 ⊆ (𝐴𝐵)))
11 df-pr 3534 . . . . . . . . . . . 12 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
12 snssi 3664 . . . . . . . . . . . . . . 15 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
13 ssun4 3242 . . . . . . . . . . . . . . 15 ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴𝐵))
1412, 13syl 14 . . . . . . . . . . . . . 14 (𝑦𝐵 → {𝑦} ⊆ (𝐴𝐵))
157, 14anim12i 336 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐵) → ({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)))
16 unss 3250 . . . . . . . . . . . . 13 (({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1715, 16sylib 121 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1811, 17eqsstrid 3143 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ⊆ (𝐴𝐵))
19 sseq1 3120 . . . . . . . . . . 11 (𝑧 = {𝑥, 𝑦} → (𝑧 ⊆ (𝐴𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴𝐵)))
2018, 19syl5ibrcom 156 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑧 = {𝑥, 𝑦} → 𝑧 ⊆ (𝐴𝐵)))
2110, 20jaod 706 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ((𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}) → 𝑧 ⊆ (𝐴𝐵)))
22 vex 2689 . . . . . . . . . 10 𝑧 ∈ V
2322elpr 3548 . . . . . . . . 9 (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
2422elpw 3516 . . . . . . . . 9 (𝑧 ∈ 𝒫 (𝐴𝐵) ↔ 𝑧 ⊆ (𝐴𝐵))
2521, 23, 243imtr4g 204 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} → 𝑧 ∈ 𝒫 (𝐴𝐵)))
2625ssrdv 3103 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
274, 26eqsstrid 3143 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵))
28 sseq1 3120 . . . . . . 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 1868 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
321, 31syl 14 . . 3 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3322elpw 3516 . . 3 (𝑧 ∈ 𝒫 𝒫 (𝐴𝐵) ↔ 𝑧 ⊆ 𝒫 (𝐴𝐵))
3432, 33sylibr 133 . 2 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴𝐵))
3534ssriv 3101 1 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ∨ wo 697   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ∪ cun 3069   ⊆ wss 3071  𝒫 cpw 3510  {csn 3527  {cpr 3528  ⟨cop 3530   × cxp 4537 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545 This theorem is referenced by:  unixpss  4652  xpexg  4653
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