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Theorem altxpsspw 25734
Description: An inclusion rule for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)
Assertion
Ref Expression
altxpsspw  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )

Proof of Theorem altxpsspw
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elaltxp 25732 . . 3  |-  ( z  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  z  =  << x ,  y >> )
2 df-altop 25715 . . . . . 6  |-  << x ,  y >>  =  { {
x } ,  {
x ,  { y } } }
3 snssi 3910 . . . . . . . . 9  |-  ( x  e.  A  ->  { x }  C_  A )
4 ssun3 3480 . . . . . . . . 9  |-  ( { x }  C_  A  ->  { x }  C_  ( A  u.  ~P B ) )
53, 4syl 16 . . . . . . . 8  |-  ( x  e.  A  ->  { x }  C_  ( A  u.  ~P B ) )
65adantr 452 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x }  C_  ( A  u.  ~P B ) )
7 elun1 3482 . . . . . . . . 9  |-  ( x  e.  A  ->  x  e.  ( A  u.  ~P B ) )
8 snssi 3910 . . . . . . . . . 10  |-  ( y  e.  B  ->  { y }  C_  B )
9 snex 4373 . . . . . . . . . . . 12  |-  { y }  e.  _V
109elpw 3773 . . . . . . . . . . 11  |-  ( { y }  e.  ~P B 
<->  { y }  C_  B )
11 elun2 3483 . . . . . . . . . . 11  |-  ( { y }  e.  ~P B  ->  { y }  e.  ( A  u.  ~P B ) )
1210, 11sylbir 205 . . . . . . . . . 10  |-  ( { y }  C_  B  ->  { y }  e.  ( A  u.  ~P B ) )
138, 12syl 16 . . . . . . . . 9  |-  ( y  e.  B  ->  { y }  e.  ( A  u.  ~P B ) )
147, 13anim12i 550 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ( A  u.  ~P B
)  /\  { y }  e.  ( A  u.  ~P B ) ) )
15 vex 2927 . . . . . . . . 9  |-  x  e. 
_V
1615, 9prss 3920 . . . . . . . 8  |-  ( ( x  e.  ( A  u.  ~P B )  /\  { y }  e.  ( A  u.  ~P B ) )  <->  { x ,  { y } }  C_  ( A  u.  ~P B ) )
1714, 16sylib 189 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x ,  {
y } }  C_  ( A  u.  ~P B ) )
18 prex 4374 . . . . . . . . 9  |-  { {
x } ,  {
x ,  { y } } }  e.  _V
1918elpw 3773 . . . . . . . 8  |-  ( { { x } ,  { x ,  {
y } } }  e.  ~P ~P ( A  u.  ~P B )  <->  { { x } ,  { x ,  {
y } } }  C_ 
~P ( A  u.  ~P B ) )
20 snex 4373 . . . . . . . . 9  |-  { x }  e.  _V
21 prex 4374 . . . . . . . . 9  |-  { x ,  { y } }  e.  _V
2220, 21prsspw 3939 . . . . . . . 8  |-  ( { { x } ,  { x ,  {
y } } }  C_ 
~P ( A  u.  ~P B )  <->  ( {
x }  C_  ( A  u.  ~P B
)  /\  { x ,  { y } }  C_  ( A  u.  ~P B ) ) )
2319, 22bitri 241 . . . . . . 7  |-  ( { { x } ,  { x ,  {
y } } }  e.  ~P ~P ( A  u.  ~P B )  <-> 
( { x }  C_  ( A  u.  ~P B )  /\  {
x ,  { y } }  C_  ( A  u.  ~P B
) ) )
246, 17, 23sylanbrc 646 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { { x } ,  { x ,  {
y } } }  e.  ~P ~P ( A  u.  ~P B ) )
252, 24syl5eqel 2496 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<< x ,  y >>  e. 
~P ~P ( A  u.  ~P B ) )
26 eleq1a 2481 . . . . 5  |-  ( << x ,  y >>  e.  ~P ~P ( A  u.  ~P B )  ->  (
z  =  << x ,  y >>  ->  z  e.  ~P ~P ( A  u.  ~P B ) ) )
2725, 26syl 16 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  << x ,  y >>  ->  z  e.  ~P ~P ( A  u.  ~P B ) ) )
2827rexlimivv 2803 . . 3  |-  ( E. x  e.  A  E. y  e.  B  z  =  << x ,  y
>>  ->  z  e.  ~P ~P ( A  u.  ~P B ) )
291, 28sylbi 188 . 2  |-  ( z  e.  ( A  XX.  B )  ->  z  e.  ~P ~P ( A  u.  ~P B ) )
3029ssriv 3320 1  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2675    u. cun 3286    C_ wss 3288   ~Pcpw 3767   {csn 3782   {cpr 3783   <<caltop 25713    XX. caltxp 25714
This theorem is referenced by:  altxpexg  25735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-pw 3769  df-sn 3788  df-pr 3789  df-altop 25715  df-altxp 25716
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