Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  altxpexg Structured version   Visualization version   GIF version

Theorem altxpexg 32391
Description: The alternate Cartesian product of two sets is a set. (Contributed by Scott Fenton, 24-Mar-2012.)
Assertion
Ref Expression
altxpexg ((𝐴𝑉𝐵𝑊) → (𝐴 ×× 𝐵) ∈ V)

Proof of Theorem altxpexg
StepHypRef Expression
1 altxpsspw 32390 . 2 (𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵)
2 pwexg 4999 . . . 4 (𝐵𝑊 → 𝒫 𝐵 ∈ V)
3 unexg 7124 . . . 4 ((𝐴𝑉 ∧ 𝒫 𝐵 ∈ V) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
42, 3sylan2 492 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 ∪ 𝒫 𝐵) ∈ V)
5 pwexg 4999 . . 3 ((𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
6 pwexg 4999 . . 3 (𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V → 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
74, 5, 63syl 18 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V)
8 ssexg 4956 . 2 (((𝐴 ×× 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∧ 𝒫 𝒫 (𝐴 ∪ 𝒫 𝐵) ∈ V) → (𝐴 ×× 𝐵) ∈ V)
91, 7, 8sylancr 698 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ×× 𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  Vcvv 3340  cun 3713  wss 3715  𝒫 cpw 4302   ×× caltxp 32370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-pw 4304  df-sn 4322  df-pr 4324  df-uni 4589  df-altop 32371  df-altxp 32372
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator