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

Theorem uniopel 4452
Description: Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opthw.1  |-  A  e. 
_V
opthw.2  |-  B  e. 
_V
Assertion
Ref Expression
uniopel  |-  ( <. A ,  B >.  e.  C  ->  U. <. A ,  B >.  e.  U. C
)

Proof of Theorem uniopel
StepHypRef Expression
1 opthw.1 . . . 4  |-  A  e. 
_V
2 opthw.2 . . . 4  |-  B  e. 
_V
31, 2uniop 4451 . . 3  |-  U. <. A ,  B >.  =  { A ,  B }
41, 2opi2 4423 . . 3  |-  { A ,  B }  e.  <. A ,  B >.
53, 4eqeltri 2505 . 2  |-  U. <. A ,  B >.  e.  <. A ,  B >.
6 elssuni 4035 . . 3  |-  ( <. A ,  B >.  e.  C  ->  <. A ,  B >.  C_  U. C )
76sseld 3339 . 2  |-  ( <. A ,  B >.  e.  C  ->  ( U. <. A ,  B >.  e. 
<. A ,  B >.  ->  U. <. A ,  B >.  e.  U. C ) )
85, 7mpi 17 1  |-  ( <. A ,  B >.  e.  C  ->  U. <. A ,  B >.  e.  U. C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   _Vcvv 2948   {cpr 3807   <.cop 3809   U.cuni 4007
This theorem is referenced by:  dmrnssfld  5121  unielrel  5386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008
  Copyright terms: Public domain W3C validator