Theorem tskxpss 9594

Description: A Cartesian product of two parts of a Tarski 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.

Step	Hyp	Ref	Expression
1		elxp2 5132	. . . . 5 |- ( z e. ( T X. T ) <-> E. x e. T E. y e. T z = <. x , y >. )
2		tskop 9593	. . . . . . . 8 |- ( ( T e. Tarski /\ x e. T /\ y e. T ) -> <. x , y >. e. T )
3		eleq1a 2696	. . . . . . . 8 |- ( <. x , y >. e. T -> ( z = <. x , y >. -> z e. T ) )
4	2, 3	syl 17	. . . . . . 7 |- ( ( T e. Tarski /\ x e. T /\ y e. T ) -> ( z = <. x , y >. -> z e. T ) )
5	4	3expib 1268	. . . . . 6 |- ( T e. Tarski -> ( ( x e. T /\ y e. T ) -> ( z = <. x , y >. -> z e. T ) ) )
6	5	rexlimdvv 3037	. . . . 5 |- ( T e. Tarski -> ( E. x e. T E. y e. T z = <. x , y >. -> z e. T ) )
7	1, 6	syl5bi 232	. . . 4 |- ( T e. Tarski -> ( z e. ( T X. T ) -> z e. T ) )
8	7	ssrdv 3609	. . 3 |- ( T e. Tarski -> ( T X. T ) C_ T )
9		xpss12 5225	. . 3 |- ( ( A C_ T /\ B C_ T ) -> ( A X. B ) C_ ( T X. T ) )
10		sstr 3611	. . . 4 |- ( ( ( A X. B ) C_ ( T X. T ) /\ ( T X. T ) C_ T ) -> ( A X. B ) C_ T )
11	10	expcom 451	. . 3 |- ( ( T X. T ) C_ T -> ( ( A X. B ) C_ ( T X. T ) -> ( A X. B ) C_ T ) )
12	8, 9, 11	syl2im 40	. 2 |- ( T e. Tarski -> ( ( A C_ T /\ B C_ T ) -> ( A X. B ) C_ T ) )
13	12	3impib 1262	1 |- ( ( T e. Tarski /\ A C_ T /\ B C_ T ) -> ( A X. B ) C_ T )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   <.cop 4183   X. cxp 5112   Tarskictsk 9570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-r1 8627  df-tsk 9571

This theorem is referenced by:  tskcard  9603