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Theorem altxpsspw 25827
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 25825 . . 3  |-  ( z  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  z  =  << x ,  y >> )
2 df-altop 25808 . . . . . 6  |-  << x ,  y >>  =  { {
x } ,  {
x ,  { y } } }
3 snssi 3944 . . . . . . . . 9  |-  ( x  e.  A  ->  { x }  C_  A )
4 ssun3 3514 . . . . . . . . 9  |-  ( { x }  C_  A  ->  { x }  C_  ( A  u.  ~P B ) )
53, 4syl 16 . . . . . . . 8  |-  ( x  e.  A  ->  { x }  C_  ( A  u.  ~P B ) )
65adantr 453 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x }  C_  ( A  u.  ~P B ) )
7 elun1 3516 . . . . . . . . 9  |-  ( x  e.  A  ->  x  e.  ( A  u.  ~P B ) )
8 snssi 3944 . . . . . . . . . 10  |-  ( y  e.  B  ->  { y }  C_  B )
9 snex 4408 . . . . . . . . . . . 12  |-  { y }  e.  _V
109elpw 3807 . . . . . . . . . . 11  |-  ( { y }  e.  ~P B 
<->  { y }  C_  B )
11 elun2 3517 . . . . . . . . . . 11  |-  ( { y }  e.  ~P B  ->  { y }  e.  ( A  u.  ~P B ) )
1210, 11sylbir 206 . . . . . . . . . 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 551 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ( A  u.  ~P B
)  /\  { y }  e.  ( A  u.  ~P B ) ) )
15 vex 2961 . . . . . . . . 9  |-  x  e. 
_V
1615, 9prss 3954 . . . . . . . 8  |-  ( ( x  e.  ( A  u.  ~P B )  /\  { y }  e.  ( A  u.  ~P B ) )  <->  { x ,  { y } }  C_  ( A  u.  ~P B ) )
1714, 16sylib 190 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x ,  {
y } }  C_  ( A  u.  ~P B ) )
18 prex 4409 . . . . . . . . 9  |-  { {
x } ,  {
x ,  { y } } }  e.  _V
1918elpw 3807 . . . . . . . 8  |-  ( { { x } ,  { x ,  {
y } } }  e.  ~P ~P ( A  u.  ~P B )  <->  { { x } ,  { x ,  {
y } } }  C_ 
~P ( A  u.  ~P B ) )
20 snex 4408 . . . . . . . . 9  |-  { x }  e.  _V
21 prex 4409 . . . . . . . . 9  |-  { x ,  { y } }  e.  _V
2220, 21prsspw 3973 . . . . . . . 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 242 . . . . . . 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 647 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { { x } ,  { x ,  {
y } } }  e.  ~P ~P ( A  u.  ~P B ) )
252, 24syl5eqel 2522 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<< x ,  y >>  e. 
~P ~P ( A  u.  ~P B ) )
26 eleq1a 2507 . . . . 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 2837 . . 3  |-  ( E. x  e.  A  E. y  e.  B  z  =  << x ,  y
>>  ->  z  e.  ~P ~P ( A  u.  ~P B ) )
291, 28sylbi 189 . 2  |-  ( z  e.  ( A  XX.  B )  ->  z  e.  ~P ~P ( A  u.  ~P B ) )
3029ssriv 3354 1  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    u. cun 3320    C_ wss 3322   ~Pcpw 3801   {csn 3816   {cpr 3817   <<caltop 25806    XX. caltxp 25807
This theorem is referenced by:  altxpexg  25828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822  df-pr 3823  df-altop 25808  df-altxp 25809
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