MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelxp2 Structured version   Unicode version

Theorem opelxp2 4860
Description: The second member of an ordered pair of classes in a Cartesian product belongs to second Cartesian product argument. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opelxp2  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  ->  B  e.  D )

Proof of Theorem opelxp2
StepHypRef Expression
1 opelxp 4856 . 2  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  <->  ( A  e.  C  /\  B  e.  D ) )
21simprbi 461 1  |-  ( <. A ,  B >.  e.  ( C  X.  D
)  ->  B  e.  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1755   <.cop 3871    X. cxp 4825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-opab 4339  df-xp 4833
This theorem is referenced by:  dff4  5845  eceqoveq  7193  isfin4-3  8472  axdc4lem  8612  canthp1lem2  8807  txcmplem1  19055  txlm  19062  brcgr  22968  nvex  23811  prsrn  26198  pprodss4v  27761
  Copyright terms: Public domain W3C validator