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Theorem altxpsspw 28006
Description: An inclusion rule for alternate Cartesian 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 28004 . . 3  |-  ( z  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  z  =  << x ,  y >> )
2 df-altop 27987 . . . . . 6  |-  << x ,  y >>  =  { {
x } ,  {
x ,  { y } } }
3 snssi 4015 . . . . . . . . 9  |-  ( x  e.  A  ->  { x }  C_  A )
4 ssun3 3519 . . . . . . . . 9  |-  ( { x }  C_  A  ->  { x }  C_  ( A  u.  ~P B ) )
53, 4syl 16 . . . . . . . 8  |-  ( x  e.  A  ->  { x }  C_  ( A  u.  ~P B ) )
65adantr 465 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x }  C_  ( A  u.  ~P B ) )
7 elun1 3521 . . . . . . . . 9  |-  ( x  e.  A  ->  x  e.  ( A  u.  ~P B ) )
8 snssi 4015 . . . . . . . . . 10  |-  ( y  e.  B  ->  { y }  C_  B )
9 snex 4531 . . . . . . . . . . . 12  |-  { y }  e.  _V
109elpw 3864 . . . . . . . . . . 11  |-  ( { y }  e.  ~P B 
<->  { y }  C_  B )
11 elun2 3522 . . . . . . . . . . 11  |-  ( { y }  e.  ~P B  ->  { y }  e.  ( A  u.  ~P B ) )
1210, 11sylbir 213 . . . . . . . . . 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 566 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( x  e.  ( A  u.  ~P B
)  /\  { y }  e.  ( A  u.  ~P B ) ) )
15 vex 2973 . . . . . . . . 9  |-  x  e. 
_V
1615, 9prss 4025 . . . . . . . 8  |-  ( ( x  e.  ( A  u.  ~P B )  /\  { y }  e.  ( A  u.  ~P B ) )  <->  { x ,  { y } }  C_  ( A  u.  ~P B ) )
1714, 16sylib 196 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x ,  {
y } }  C_  ( A  u.  ~P B ) )
18 prex 4532 . . . . . . . . 9  |-  { {
x } ,  {
x ,  { y } } }  e.  _V
1918elpw 3864 . . . . . . . 8  |-  ( { { x } ,  { x ,  {
y } } }  e.  ~P ~P ( A  u.  ~P B )  <->  { { x } ,  { x ,  {
y } } }  C_ 
~P ( A  u.  ~P B ) )
20 snex 4531 . . . . . . . . 9  |-  { x }  e.  _V
21 prex 4532 . . . . . . . . 9  |-  { x ,  { y } }  e.  _V
2220, 21prsspw 4043 . . . . . . . 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 249 . . . . . . 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 664 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { { x } ,  { x ,  {
y } } }  e.  ~P ~P ( A  u.  ~P B ) )
252, 24syl5eqel 2525 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<< x ,  y >>  e. 
~P ~P ( A  u.  ~P B ) )
26 eleq1a 2510 . . . . 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 2844 . . 3  |-  ( E. x  e.  A  E. y  e.  B  z  =  << x ,  y
>>  ->  z  e.  ~P ~P ( A  u.  ~P B ) )
291, 28sylbi 195 . 2  |-  ( z  e.  ( A  XX.  B )  ->  z  e.  ~P ~P ( A  u.  ~P B ) )
3029ssriv 3358 1  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2714    u. cun 3324    C_ wss 3326   ~Pcpw 3858   {csn 3875   {cpr 3877   <<caltop 27985    XX. caltxp 27986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-pw 3860  df-sn 3876  df-pr 3878  df-altop 27987  df-altxp 27988
This theorem is referenced by:  altxpexg  28007
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