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

Theorem altxpsspw 29201
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 29199 . . 3  |-  ( z  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  z  =  << x ,  y >> )
2 df-altop 29182 . . . . . 6  |-  << x ,  y >>  =  { {
x } ,  {
x ,  { y } } }
3 snssi 4171 . . . . . . . . 9  |-  ( x  e.  A  ->  { x }  C_  A )
4 ssun3 3669 . . . . . . . . 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 3671 . . . . . . . . 9  |-  ( x  e.  A  ->  x  e.  ( A  u.  ~P B ) )
8 snssi 4171 . . . . . . . . . 10  |-  ( y  e.  B  ->  { y }  C_  B )
9 snex 4688 . . . . . . . . . . . 12  |-  { y }  e.  _V
109elpw 4016 . . . . . . . . . . 11  |-  ( { y }  e.  ~P B 
<->  { y }  C_  B )
11 elun2 3672 . . . . . . . . . . 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 3116 . . . . . . . . 9  |-  x  e. 
_V
1615, 9prss 4181 . . . . . . . 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 4689 . . . . . . . . 9  |-  { {
x } ,  {
x ,  { y } } }  e.  _V
1918elpw 4016 . . . . . . . 8  |-  ( { { x } ,  { x ,  {
y } } }  e.  ~P ~P ( A  u.  ~P B )  <->  { { x } ,  { x ,  {
y } } }  C_ 
~P ( A  u.  ~P B ) )
20 snex 4688 . . . . . . . . 9  |-  { x }  e.  _V
21 prex 4689 . . . . . . . . 9  |-  { x ,  { y } }  e.  _V
2220, 21prsspw 4199 . . . . . . . 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 2559 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<< x ,  y >>  e. 
~P ~P ( A  u.  ~P B ) )
26 eleq1a 2550 . . . . 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 2960 . . 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 3508 1  |-  ( A 
XX.  B )  C_  ~P ~P ( A  u.  ~P B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815    u. cun 3474    C_ wss 3476   ~Pcpw 4010   {csn 4027   {cpr 4029   <<caltop 29180    XX. caltxp 29181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-pw 4012  df-sn 4028  df-pr 4030  df-altop 29182  df-altxp 29183
This theorem is referenced by:  altxpexg  29202
  Copyright terms: Public domain W3C validator