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Theorem altxpsspw 24583
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 24581 . . 3  |-  ( z  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  z  =  << x ,  y >> )
2 df-altop 24564 . . . . . 6  |-  << x ,  y >>  =  { {
x } ,  {
x ,  { y } } }
3 snssi 3775 . . . . . . . . 9  |-  ( x  e.  A  ->  { x }  C_  A )
4 ssun3 3353 . . . . . . . . 9  |-  ( { x }  C_  A  ->  { x }  C_  ( A  u.  ~P B ) )
53, 4syl 15 . . . . . . . 8  |-  ( x  e.  A  ->  { x }  C_  ( A  u.  ~P B ) )
65adantr 451 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x }  C_  ( A  u.  ~P B ) )
7 elun1 3355 . . . . . . . . 9  |-  ( x  e.  A  ->  x  e.  ( A  u.  ~P B ) )
8 snssi 3775 . . . . . . . . . 10  |-  ( y  e.  B  ->  { y }  C_  B )
9 snex 4232 . . . . . . . . . . . 12  |-  { y }  e.  _V
109elpw 3644 . . . . . . . . . . 11  |-  ( { y }  e.  ~P B 
<->  { y }  C_  B )
11 elun2 3356 . . . . . . . . . . 11  |-  ( { y }  e.  ~P B  ->  { y }  e.  ( A  u.  ~P B ) )
1210, 11sylbir 204 . . . . . . . . . 10  |-  ( { y }  C_  B  ->  { y }  e.  ( A  u.  ~P B ) )
138, 12syl 15 . . . . . . . . 9  |-  ( y  e.  B  ->  { y }  e.  ( A  u.  ~P B ) )
147, 13anim12i 549 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ( A  u.  ~P B
)  /\  { y }  e.  ( A  u.  ~P B ) ) )
15 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
1615, 9prss 3785 . . . . . . . 8  |-  ( ( x  e.  ( A  u.  ~P B )  /\  { y }  e.  ( A  u.  ~P B ) )  <->  { x ,  { y } }  C_  ( A  u.  ~P B ) )
1714, 16sylib 188 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x ,  {
y } }  C_  ( A  u.  ~P B ) )
18 prex 4233 . . . . . . . . 9  |-  { {
x } ,  {
x ,  { y } } }  e.  _V
1918elpw 3644 . . . . . . . 8  |-  ( { { x } ,  { x ,  {
y } } }  e.  ~P ~P ( A  u.  ~P B )  <->  { { x } ,  { x ,  {
y } } }  C_ 
~P ( A  u.  ~P B ) )
20 snex 4232 . . . . . . . . 9  |-  { x }  e.  _V
21 prex 4233 . . . . . . . . 9  |-  { x ,  { y } }  e.  _V
2220, 21prsspw 3801 . . . . . . . 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 240 . . . . . . 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 645 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { { x } ,  { x ,  {
y } } }  e.  ~P ~P ( A  u.  ~P B ) )
252, 24syl5eqel 2380 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<< x ,  y >>  e. 
~P ~P ( A  u.  ~P B ) )
26 eleq1a 2365 . . . . 5  |-  ( << x ,  y >>  e.  ~P ~P ( A  u.  ~P B )  ->  (
z  =  << x ,  y >>  ->  z  e.  ~P ~P ( A  u.  ~P B ) ) )
2725, 26syl 15 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( z  =  << x ,  y >>  ->  z  e.  ~P ~P ( A  u.  ~P B ) ) )
2827rexlimivv 2685 . . 3  |-  ( E. x  e.  A  E. y  e.  B  z  =  << x ,  y
>>  ->  z  e.  ~P ~P ( A  u.  ~P B ) )
291, 28sylbi 187 . 2  |-  ( z  e.  ( A  XX.  B )  ->  z  e.  ~P ~P ( A  u.  ~P B ) )
3029ssriv 3197 1  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    u. cun 3163    C_ wss 3165   ~Pcpw 3638   {csn 3653   {cpr 3654   <<caltop 24562    XX. caltxp 24563
This theorem is referenced by:  altxpexg  24584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-pw 3640  df-sn 3659  df-pr 3660  df-altop 24564  df-altxp 24565
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