Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  altopth Structured version   Unicode version

Theorem altopth 30280
Description: The alternate ordered pair theorem. If two alternate ordered pairs are equal, their first elements are equal and their second elements are equal. Note that  C and  D are not required to be a set due to a peculiarity of our specific ordered pair definition, as opposed to the regular ordered pairs used here, which (as in opth 4664), requires  D to be a set. (Contributed by Scott Fenton, 23-Mar-2012.)
Hypotheses
Ref Expression
altopth.1  |-  A  e. 
_V
altopth.2  |-  B  e. 
_V
Assertion
Ref Expression
altopth  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem altopth
StepHypRef Expression
1 altopth.1 . 2  |-  A  e. 
_V
2 altopth.2 . 2  |-  B  e. 
_V
3 altopthg 30278 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D )
) )
41, 2, 3mp2an 670 1  |-  ( << A ,  B >>  =  << C ,  D >>  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058   <<caltop 30267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-sn 3972  df-pr 3974  df-altop 30269
This theorem is referenced by:  altopthd  30283  altopelaltxp  30287
  Copyright terms: Public domain W3C validator