ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpsspw GIF version

Theorem xpsspw 4498
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 4407 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
2 vex 2613 . . . . . . . 8 𝑥 ∈ V
3 vex 2613 . . . . . . . 8 𝑦 ∈ V
42, 3dfop 3589 . . . . . . 7 𝑥, 𝑦⟩ = {{𝑥}, {𝑥, 𝑦}}
5 snssi 3549 . . . . . . . . . . . . 13 (𝑥𝐴 → {𝑥} ⊆ 𝐴)
6 ssun3 3147 . . . . . . . . . . . . 13 ({𝑥} ⊆ 𝐴 → {𝑥} ⊆ (𝐴𝐵))
75, 6syl 14 . . . . . . . . . . . 12 (𝑥𝐴 → {𝑥} ⊆ (𝐴𝐵))
87adantr 270 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → {𝑥} ⊆ (𝐴𝐵))
9 sseq1 3029 . . . . . . . . . . 11 (𝑧 = {𝑥} → (𝑧 ⊆ (𝐴𝐵) ↔ {𝑥} ⊆ (𝐴𝐵)))
108, 9syl5ibrcom 155 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑧 = {𝑥} → 𝑧 ⊆ (𝐴𝐵)))
11 df-pr 3423 . . . . . . . . . . . 12 {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
12 snssi 3549 . . . . . . . . . . . . . . 15 (𝑦𝐵 → {𝑦} ⊆ 𝐵)
13 ssun4 3148 . . . . . . . . . . . . . . 15 ({𝑦} ⊆ 𝐵 → {𝑦} ⊆ (𝐴𝐵))
1412, 13syl 14 . . . . . . . . . . . . . 14 (𝑦𝐵 → {𝑦} ⊆ (𝐴𝐵))
157, 14anim12i 331 . . . . . . . . . . . . 13 ((𝑥𝐴𝑦𝐵) → ({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)))
16 unss 3156 . . . . . . . . . . . . 13 (({𝑥} ⊆ (𝐴𝐵) ∧ {𝑦} ⊆ (𝐴𝐵)) ↔ ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1715, 16sylib 120 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → ({𝑥} ∪ {𝑦}) ⊆ (𝐴𝐵))
1811, 17syl5eqss 3052 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → {𝑥, 𝑦} ⊆ (𝐴𝐵))
19 sseq1 3029 . . . . . . . . . . 11 (𝑧 = {𝑥, 𝑦} → (𝑧 ⊆ (𝐴𝐵) ↔ {𝑥, 𝑦} ⊆ (𝐴𝐵)))
2018, 19syl5ibrcom 155 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑧 = {𝑥, 𝑦} → 𝑧 ⊆ (𝐴𝐵)))
2110, 20jaod 670 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → ((𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}) → 𝑧 ⊆ (𝐴𝐵)))
22 vex 2613 . . . . . . . . . 10 𝑧 ∈ V
2322elpr 3437 . . . . . . . . 9 (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} ↔ (𝑧 = {𝑥} ∨ 𝑧 = {𝑥, 𝑦}))
2422elpw 3406 . . . . . . . . 9 (𝑧 ∈ 𝒫 (𝐴𝐵) ↔ 𝑧 ⊆ (𝐴𝐵))
2521, 23, 243imtr4g 203 . . . . . . . 8 ((𝑥𝐴𝑦𝐵) → (𝑧 ∈ {{𝑥}, {𝑥, 𝑦}} → 𝑧 ∈ 𝒫 (𝐴𝐵)))
2625ssrdv 3014 . . . . . . 7 ((𝑥𝐴𝑦𝐵) → {{𝑥}, {𝑥, 𝑦}} ⊆ 𝒫 (𝐴𝐵))
274, 26syl5eqss 3052 . . . . . 6 ((𝑥𝐴𝑦𝐵) → ⟨𝑥, 𝑦⟩ ⊆ 𝒫 (𝐴𝐵))
28 sseq1 3029 . . . . . . 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 1819 . . . 4 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
321, 31syl 14 . . 3 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ⊆ 𝒫 (𝐴𝐵))
3322elpw 3406 . . 3 (𝑧 ∈ 𝒫 𝒫 (𝐴𝐵) ↔ 𝑧 ⊆ 𝒫 (𝐴𝐵))
3432, 33sylibr 132 . 2 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 ∈ 𝒫 𝒫 (𝐴𝐵))
3534ssriv 3012 1 (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 102  wo 662   = wceq 1285  wex 1422  wcel 1434  cun 2980  wss 2982  𝒫 cpw 3400  {csn 3416  {cpr 3417  cop 3419   × cxp 4389
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860  df-xp 4397
This theorem is referenced by:  unixpss  4499  xpexg  4500
  Copyright terms: Public domain W3C validator