Theorem otel3xp 5153

Description: An ordered triple is an element of a doubled Cartesian product. (Contributed by Alexander van der Vekens, 26-Feb-2018.)

Assertion

Ref	Expression
otel3xp	|- ( ( T = <. A , B , C >. /\ ( A e. X /\ B e. Y /\ C e. Z ) ) -> T e. ( ( X X. Y ) X. Z ) )

Proof of Theorem otel3xp

Step	Hyp	Ref	Expression
1		df-ot 4186	. . . 4 |- <. A , B , C >. = <. <. A , B >. , C >.
2		3simpa 1058	. . . . . 6 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> ( A e. X /\ B e. Y ) )
3		opelxp 5146	. . . . . 6 |- ( <. A , B >. e. ( X X. Y ) <-> ( A e. X /\ B e. Y ) )
4	2, 3	sylibr 224	. . . . 5 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> <. A , B >. e. ( X X. Y ) )
5		simp3 1063	. . . . 5 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> C e. Z )
6	4, 5	opelxpd 5149	. . . 4 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> <. <. A , B >. , C >. e. ( ( X X. Y ) X. Z ) )
7	1, 6	syl5eqel 2705	. . 3 |- ( ( A e. X /\ B e. Y /\ C e. Z ) -> <. A , B , C >. e. ( ( X X. Y ) X. Z ) )
8		eleq1 2689	. . 3 |- ( T = <. A , B , C >. -> ( T e. ( ( X X. Y ) X. Z ) <-> <. A , B , C >. e. ( ( X X. Y ) X. Z ) ) )
9	7, 8	syl5ibr 236	. 2 |- ( T = <. A , B , C >. -> ( ( A e. X /\ B e. Y /\ C e. Z ) -> T e. ( ( X X. Y ) X. Z ) ) )
10	9	imp 445	1 |- ( ( T = <. A , B , C >. /\ ( A e. X /\ B e. Y /\ C e. Z ) ) -> T e. ( ( X X. Y ) X. Z ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   <.cotp 4185   X. cxp 5112

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-opab 4713  df-xp 5120

This theorem is referenced by: (None)