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Theorem xpsspwOLD 5115
Description: A Cartesian product is included in the power of the power of the union of its arguments. (Contributed by NM, 13-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
xpsspwOLD  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )

Proof of Theorem xpsspwOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5108 . 2  |-  Rel  ( A  X.  B )
2 opelxp 5028 . . 3  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
3 snssi 4171 . . . . . . . 8  |-  ( x  e.  A  ->  { x }  C_  A )
4 ssun3 3669 . . . . . . . 8  |-  ( { x }  C_  A  ->  { x }  C_  ( A  u.  B
) )
53, 4syl 16 . . . . . . 7  |-  ( x  e.  A  ->  { x }  C_  ( A  u.  B ) )
6 snex 4688 . . . . . . . 8  |-  { x }  e.  _V
76elpw 4016 . . . . . . 7  |-  ( { x }  e.  ~P ( A  u.  B
)  <->  { x }  C_  ( A  u.  B
) )
85, 7sylibr 212 . . . . . 6  |-  ( x  e.  A  ->  { x }  e.  ~P ( A  u.  B )
)
98adantr 465 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x }  e.  ~P ( A  u.  B
) )
10 df-pr 4030 . . . . . . 7  |-  { x ,  y }  =  ( { x }  u.  { y } )
11 snssi 4171 . . . . . . . . . 10  |-  ( y  e.  B  ->  { y }  C_  B )
12 ssun4 3670 . . . . . . . . . 10  |-  ( { y }  C_  B  ->  { y }  C_  ( A  u.  B
) )
1311, 12syl 16 . . . . . . . . 9  |-  ( y  e.  B  ->  { y }  C_  ( A  u.  B ) )
145, 13anim12i 566 . . . . . . . 8  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( { x }  C_  ( A  u.  B
)  /\  { y }  C_  ( A  u.  B ) ) )
15 unss 3678 . . . . . . . 8  |-  ( ( { x }  C_  ( A  u.  B
)  /\  { y }  C_  ( A  u.  B ) )  <->  ( {
x }  u.  {
y } )  C_  ( A  u.  B
) )
1614, 15sylib 196 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( { x }  u.  { y } ) 
C_  ( A  u.  B ) )
1710, 16syl5eqss 3548 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x ,  y }  C_  ( A  u.  B ) )
18 zfpair2 4687 . . . . . . 7  |-  { x ,  y }  e.  _V
1918elpw 4016 . . . . . 6  |-  ( { x ,  y }  e.  ~P ( A  u.  B )  <->  { x ,  y }  C_  ( A  u.  B
) )
2017, 19sylibr 212 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  { x ,  y }  e.  ~P ( A  u.  B )
)
219, 20jca 532 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( { x }  e.  ~P ( A  u.  B )  /\  {
x ,  y }  e.  ~P ( A  u.  B ) ) )
22 prex 4689 . . . . . 6  |-  { {
x } ,  {
x ,  y } }  e.  _V
2322elpw 4016 . . . . 5  |-  ( { { x } ,  { x ,  y } }  e.  ~P ~P ( A  u.  B
)  <->  { { x } ,  { x ,  y } }  C_  ~P ( A  u.  B
) )
24 vex 3116 . . . . . . 7  |-  x  e. 
_V
25 vex 3116 . . . . . . 7  |-  y  e. 
_V
2624, 25dfop 4212 . . . . . 6  |-  <. x ,  y >.  =  { { x } ,  { x ,  y } }
2726eleq1i 2544 . . . . 5  |-  ( <.
x ,  y >.  e.  ~P ~P ( A  u.  B )  <->  { { x } ,  { x ,  y } }  e.  ~P ~P ( A  u.  B ) )
286, 18prss 4181 . . . . 5  |-  ( ( { x }  e.  ~P ( A  u.  B
)  /\  { x ,  y }  e.  ~P ( A  u.  B
) )  <->  { { x } ,  { x ,  y } }  C_ 
~P ( A  u.  B ) )
2923, 27, 283bitr4ri 278 . . . 4  |-  ( ( { x }  e.  ~P ( A  u.  B
)  /\  { x ,  y }  e.  ~P ( A  u.  B
) )  <->  <. x ,  y >.  e.  ~P ~P ( A  u.  B
) )
3021, 29sylib 196 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  ~P ~P ( A  u.  B ) )
312, 30sylbi 195 . 2  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  ->  <. x ,  y >.  e.  ~P ~P ( A  u.  B
) )
321, 31relssi 5092 1  |-  ( A  X.  B )  C_  ~P ~P ( A  u.  B )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1767    u. cun 3474    C_ wss 3476   ~Pcpw 4010   {csn 4027   {cpr 4029   <.cop 4033    X. cxp 4997
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-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005  df-rel 5006
This theorem is referenced by: (None)
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