ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  otelxp1 Unicode version

Theorem otelxp1 4545
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 4543 . 2  |-  ( <. <. A ,  B >. ,  C >.  e.  (
( R  X.  S
)  X.  T )  ->  <. A ,  B >.  e.  ( R  X.  S ) )
2 opelxp1 4543 . 2  |-  ( <. A ,  B >.  e.  ( R  X.  S
)  ->  A  e.  R )
31, 2syl 14 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 1465   <.cop 3500    X. cxp 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-xp 4515
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator