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Theorem xpsspw 4477
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 4388 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2 vex 2577 . . . . . . . 8 𝑥 ∈ V
3 vex 2577 . . . . . . . 8 𝑦 ∈ V
42, 3dfop 3575 . . . . . . 7 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
5 snssi 3535 . . . . . . . . . . . . 13 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
6 ssun3 3135 . . . . . . . . . . . . 13 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴𝐵))
75, 6syl 14 . . . . . . . . . . . 12 (𝑥𝐴 → {𝑥} ⊆ (𝐴𝐵))
87adantr 265 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → {𝑥} ⊆ (𝐴𝐵))
9 sseq1 2993 . . . . . . . . . . 11 (𝑧 = {𝑥} → (𝑧 ⊆ (𝐴𝐵) ↔ {𝑥} ⊆ (𝐴𝐵)))
108, 9syl5ibrcom 150 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑧 = {𝑥} → 𝑧 ⊆ (𝐴𝐵)))
11 df-pr 3409 . . . . . . . . . . . 12 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
12 snssi 3535 . . . . . . . . . . . . . . 15 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
13 ssun4 3136 . . . . . . . . . . . . . . 15 ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴𝐵))
1412, 13syl 14 . . . . . . . . . . . . . 14 (𝑦𝐵 → {𝑦} ⊆ (𝐴𝐵))
157, 14anim12i 325 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐵) → ({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)))
16 unss 3144 . . . . . . . . . . . . 13 (({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1715, 16sylib 131 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1811, 17syl5eqss 3016 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ⊆ (𝐴𝐵))
19 sseq1 2993 . . . . . . . . . . 11 (𝑧 = {𝑥, 𝑦} → (𝑧 ⊆ (𝐴𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴𝐵)))
2018, 19syl5ibrcom 150 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑧 = {𝑥, 𝑦} → 𝑧 ⊆ (𝐴𝐵)))
2110, 20jaod 647 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ((𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}) → 𝑧 ⊆ (𝐴𝐵)))
22 vex 2577 . . . . . . . . . 10 𝑧 ∈ V
2322elpr 3423 . . . . . . . . 9 (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
2422elpw 3392 . . . . . . . . 9 (𝑧 ∈ 𝒫 (𝐴𝐵) ↔ 𝑧 ⊆ (𝐴𝐵))
2521, 23, 243imtr4g 198 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} → 𝑧 ∈ 𝒫 (𝐴𝐵)))
2625ssrdv 2978 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
274, 26syl5eqss 3016 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵))
28 sseq1 2993 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ⊆ 𝒫 (𝐴𝐵) ↔ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵)))
2928biimpar 285 . . . . . 6 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3027, 29sylan2 274 . . . . 5 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3130exlimivv 1792 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
321, 31syl 14 . . 3 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3322elpw 3392 . . 3 (𝑧 ∈ 𝒫 𝒫 (𝐴𝐵) ↔ 𝑧 ⊆ 𝒫 (𝐴𝐵))
3432, 33sylibr 141 . 2 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴𝐵))
3534ssriv 2976 1 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 101  wo 639   = wceq 1259  wex 1397  wcel 1409  cun 2942  wss 2944  𝒫 cpw 3386  {csn 3402  {cpr 3403  cop 3405   × cxp 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-xp 4378
This theorem is referenced by:  unixpss  4478  xpexg  4479
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