Theorem opth2 4375

Description: Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)

Hypotheses

Ref	Expression
opth2.1	|- C e. _V
opth2.2	|- D e. _V

Assertion

Ref	Expression
opth2	|- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )

Proof of Theorem opth2

Step	Hyp	Ref	Expression
1		opth2.1	. 2 |- C e. _V
2		opth2.2	. 2 |- D e. _V
3		opthg2 4374	. 2 |- ( ( C e. _V /\ D e. _V ) -> ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) ) )
4	1, 2, 3	mp2an 430	1 |- ( <. A , B >. = <. C , D >. <-> ( A = C /\ B = D ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   <.cop 3708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714

This theorem is referenced by:  eqvinop  4378  opelxp  4799  fsn  5871  dfplpq2  7711  ltresr  8196  frecuzrdgtcl  10827  frecuzrdgfunlem  10834