MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  otelxp1 Unicode version

Theorem otelxp1 4855
Description: The first member of an ordered triple of classes in a cross product belongs to first cross product argument. (Contributed by NM, 28-May-2008.)
Assertion
Ref Expression
otelxp1  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( R  X.  S
)  X.  T )  ->  A  e.  R
)

Proof of Theorem otelxp1
StepHypRef Expression
1 opelxp1 4853 . 2  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( R  X.  S
)  X.  T )  ->  <. A ,  B >.  e.  ( R  X.  S ) )
2 opelxp1 4853 . 2  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  ->  A  e.  R )
31, 2syl 16 1  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( R  X.  S
)  X.  T )  ->  A  e.  R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   <.cop 3762    X. cxp 4818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-opab 4210  df-xp 4826
  Copyright terms: Public domain W3C validator