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Theorem tskxpss 8394
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 4707 . . . . 5  |-  ( z  e.  ( T  X.  T )  <->  E. x  e.  T  E. y  e.  T  z  =  <. x ,  y >.
)
2 tskop 8393 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  e.  T )  ->  <. x ,  y >.  e.  T
)
3 eleq1a 2352 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  T  ->  ( z  =  <. x ,  y
>.  ->  z  e.  T
) )
42, 3syl 15 . . . . . . 7  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  e.  T )  ->  (
z  =  <. x ,  y >.  ->  z  e.  T ) )
543expib 1154 . . . . . 6  |-  ( T  e.  Tarski  ->  ( ( x  e.  T  /\  y  e.  T )  ->  (
z  =  <. x ,  y >.  ->  z  e.  T ) ) )
65rexlimdvv 2673 . . . . 5  |-  ( T  e.  Tarski  ->  ( E. x  e.  T  E. y  e.  T  z  =  <. x ,  y >.  ->  z  e.  T ) )
71, 6syl5bi 208 . . . 4  |-  ( T  e.  Tarski  ->  ( z  e.  ( T  X.  T
)  ->  z  e.  T ) )
87ssrdv 3185 . . 3  |-  ( T  e.  Tarski  ->  ( T  X.  T )  C_  T
)
9 xpss12 4792 . . 3  |-  ( ( A  C_  T  /\  B  C_  T )  -> 
( A  X.  B
)  C_  ( T  X.  T ) )
10 sstr 3187 . . . 4  |-  ( ( ( A  X.  B
)  C_  ( T  X.  T )  /\  ( T  X.  T )  C_  T )  ->  ( A  X.  B )  C_  T )
1110expcom 424 . . 3  |-  ( ( T  X.  T ) 
C_  T  ->  (
( A  X.  B
)  C_  ( T  X.  T )  ->  ( A  X.  B )  C_  T ) )
128, 9, 11syl2im 34 . 2  |-  ( T  e.  Tarski  ->  ( ( A 
C_  T  /\  B  C_  T )  ->  ( A  X.  B )  C_  T ) )
13123impib 1149 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   <.cop 3643    X. cxp 4687   Tarskictsk 8370
This theorem is referenced by:  tskcard  8403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-r1 7436  df-tsk 8371
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