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Theorem opthneg 4583
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 2620 . 2  |-  ( <. A ,  B >.  =/= 
<. C ,  D >.  <->  -.  <. A ,  B >.  = 
<. C ,  D >. )
2 opthg 4579 . . . 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 2620 . . . . . . 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 2620 . . . . . . 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 1369    e. wcel 1756    =/= wne 2618   <.cop 3895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896
This theorem is referenced by:  opthne  4584  zlmodzxznm  31051
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