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

Step	Hyp	Ref	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 ) ) )
4	1, 2, 3	mp2an 670	1 |- ( << A , B >> = << C , D >> <-> ( A = C /\ B = D ) )

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