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Theorem tskxpss 8481
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 4786 . . . . 5  |-  ( z  e.  ( T  X.  T )  <->  E. x  e.  T  E. y  e.  T  z  =  <. x ,  y >.
)
2 tskop 8480 . . . . . . . 8  |-  ( ( T  e.  Tarski  /\  x  e.  T  /\  y  e.  T )  ->  <. x ,  y >.  e.  T
)
3 eleq1a 2427 . . . . . . . 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 2749 . . . . 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 3261 . . 3  |-  ( T  e.  Tarski  ->  ( T  X.  T )  C_  T
)
9 xpss12 4871 . . 3  |-  ( ( A  C_  T  /\  B  C_  T )  -> 
( A  X.  B
)  C_  ( T  X.  T ) )
10 sstr 3263 . . . 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 1642    e. wcel 1710   E.wrex 2620    C_ wss 3228   <.cop 3719    X. cxp 4766   Tarskictsk 8457
This theorem is referenced by:  tskcard  8490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-r1 7523  df-tsk 8458
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