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Theorem tskxpss 8652
Description: A cross product of two parts of a Tarski's class is a part of the class. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Mario Carneiro, 20-Jun-2013.)
Assertion
Ref Expression
tskxpss  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  B  C_  T )  ->  ( A  X.  B )  C_  T )

Proof of Theorem tskxpss
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 4899 . . . . 5  |-  ( z  e.  ( T  X.  T )  <->  E. x  e.  T  E. y  e.  T  z  =  <. x ,  y >.
)
2 tskop 8651 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  e.  T )  ->  <. x ,  y >.  e.  T
)
3 eleq1a 2507 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  T  ->  ( z  =  <. x ,  y
>.  ->  z  e.  T
) )
42, 3syl 16 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  e.  T )  ->  (
z  =  <. x ,  y >.  ->  z  e.  T ) )
543expib 1157 . . . . . 6  |-  ( T  e.  Tarski  ->  ( ( x  e.  T  /\  y  e.  T )  ->  (
z  =  <. x ,  y >.  ->  z  e.  T ) ) )
65rexlimdvv 2838 . . . . 5  |-  ( T  e.  Tarski  ->  ( E. x  e.  T  E. y  e.  T  z  =  <. x ,  y >.  ->  z  e.  T ) )
71, 6syl5bi 210 . . . 4  |-  ( T  e.  Tarski  ->  ( z  e.  ( T  X.  T
)  ->  z  e.  T ) )
87ssrdv 3356 . . 3  |-  ( T  e.  Tarski  ->  ( T  X.  T )  C_  T
)
9 xpss12 4984 . . 3  |-  ( ( A  C_  T  /\  B  C_  T )  -> 
( A  X.  B
)  C_  ( T  X.  T ) )
10 sstr 3358 . . . 4  |-  ( ( ( A  X.  B
)  C_  ( T  X.  T )  /\  ( T  X.  T )  C_  T )  ->  ( A  X.  B )  C_  T )
1110expcom 426 . . 3  |-  ( ( T  X.  T ) 
C_  T  ->  (
( A  X.  B
)  C_  ( T  X.  T )  ->  ( A  X.  B )  C_  T ) )
128, 9, 11syl2im 37 . 2  |-  ( T  e.  Tarski  ->  ( ( A 
C_  T  /\  B  C_  T )  ->  ( A  X.  B )  C_  T ) )
13123impib 1152 1  |-  ( ( T  e.  Tarski  /\  A  C_  T  /\  B  C_  T )  ->  ( A  X.  B )  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708    C_ wss 3322   <.cop 3819    X. cxp 4879   Tarskictsk 8628
This theorem is referenced by:  tskcard  8661
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-r1 7693  df-tsk 8629
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