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

Theorem opthneg 4726
Description: Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018.)
Assertion
Ref Expression
opthneg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =/=  <. C ,  D >.  <-> 
( A  =/=  C  \/  B  =/=  D
) ) )

Proof of Theorem opthneg
StepHypRef Expression
1 df-ne 2664 . 2  |-  ( <. A ,  B >.  =/= 
<. C ,  D >.  <->  -.  <. A ,  B >.  = 
<. C ,  D >. )
2 opthg 4722 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )
32notbid 294 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  <. A ,  B >.  =  <. C ,  D >. 
<->  -.  ( A  =  C  /\  B  =  D ) ) )
4 ianor 488 . . . 4  |-  ( -.  ( A  =  C  /\  B  =  D )  <->  ( -.  A  =  C  \/  -.  B  =  D )
)
5 df-ne 2664 . . . . . . 7  |-  ( A  =/=  C  <->  -.  A  =  C )
65bicomi 202 . . . . . 6  |-  ( -.  A  =  C  <->  A  =/=  C )
76a1i 11 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  A  =  C  <->  A  =/=  C
) )
8 df-ne 2664 . . . . . . 7  |-  ( B  =/=  D  <->  -.  B  =  D )
98bicomi 202 . . . . . 6  |-  ( -.  B  =  D  <->  B  =/=  D )
109a1i 11 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  B  =  D  <->  B  =/=  D
) )
117, 10orbi12d 709 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( -.  A  =  C  \/  -.  B  =  D )  <->  ( A  =/=  C  \/  B  =/=  D ) ) )
124, 11syl5bb 257 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  ( A  =  C  /\  B  =  D )  <->  ( A  =/=  C  \/  B  =/= 
D ) ) )
133, 12bitrd 253 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( -.  <. A ,  B >.  =  <. C ,  D >. 
<->  ( A  =/=  C  \/  B  =/=  D
) ) )
141, 13syl5bb 257 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =/=  <. C ,  D >.  <-> 
( A  =/=  C  \/  B  =/=  D
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   <.cop 4033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034
This theorem is referenced by:  opthne  4727  zlmodzxznm  32396
  Copyright terms: Public domain W3C validator