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Theorem isso2i 4673
Description: Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
isso2i.1  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <->  -.  ( x  =  y  \/  y R x ) ) )
isso2i.2  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( ( x R y  /\  y R z )  ->  x R z ) )
Assertion
Ref Expression
isso2i  |-  R  Or  A
Distinct variable groups:    x, y,
z, R    x, A, y, z

Proof of Theorem isso2i
StepHypRef Expression
1 equid 1729 . . . . 5  |-  x  =  x
21orci 390 . . . 4  |-  ( x  =  x  \/  x R x )
3 eleq1 2503 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  A  <->  x  e.  A ) )
43anbi2d 703 . . . . . 6  |-  ( y  =  x  ->  (
( x  e.  A  /\  y  e.  A
)  <->  ( x  e.  A  /\  x  e.  A ) ) )
5 equequ2 1737 . . . . . . . 8  |-  ( y  =  x  ->  (
x  =  y  <->  x  =  x ) )
6 breq1 4295 . . . . . . . 8  |-  ( y  =  x  ->  (
y R x  <->  x R x ) )
75, 6orbi12d 709 . . . . . . 7  |-  ( y  =  x  ->  (
( x  =  y  \/  y R x )  <->  ( x  =  x  \/  x R x ) ) )
8 breq2 4296 . . . . . . . 8  |-  ( y  =  x  ->  (
x R y  <->  x R x ) )
98notbid 294 . . . . . . 7  |-  ( y  =  x  ->  ( -.  x R y  <->  -.  x R x ) )
107, 9bibi12d 321 . . . . . 6  |-  ( y  =  x  ->  (
( ( x  =  y  \/  y R x )  <->  -.  x R y )  <->  ( (
x  =  x  \/  x R x )  <->  -.  x R x ) ) )
114, 10imbi12d 320 . . . . 5  |-  ( y  =  x  ->  (
( ( x  e.  A  /\  y  e.  A )  ->  (
( x  =  y  \/  y R x )  <->  -.  x R
y ) )  <->  ( (
x  e.  A  /\  x  e.  A )  ->  ( ( x  =  x  \/  x R x )  <->  -.  x R x ) ) ) )
12 isso2i.1 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  <->  -.  ( x  =  y  \/  y R x ) ) )
1312con2bid 329 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( x  =  y  \/  y R x )  <->  -.  x R y ) )
1411, 13chvarv 1958 . . . 4  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( ( x  =  x  \/  x R x )  <->  -.  x R x ) )
152, 14mpbii 211 . . 3  |-  ( ( x  e.  A  /\  x  e.  A )  ->  -.  x R x )
1615anidms 645 . 2  |-  ( x  e.  A  ->  -.  x R x )
17 isso2i.2 . 2  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( ( x R y  /\  y R z )  ->  x R z ) )
1813biimprd 223 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( -.  x R y  ->  ( x  =  y  \/  y R x ) ) )
19 3orass 968 . . . 4  |-  ( ( x R y  \/  x  =  y  \/  y R x )  <-> 
( x R y  \/  ( x  =  y  \/  y R x ) ) )
20 df-or 370 . . . 4  |-  ( ( x R y  \/  ( x  =  y  \/  y R x ) )  <->  ( -.  x R y  ->  (
x  =  y  \/  y R x ) ) )
2119, 20bitri 249 . . 3  |-  ( ( x R y  \/  x  =  y  \/  y R x )  <-> 
( -.  x R y  ->  ( x  =  y  \/  y R x ) ) )
2218, 21sylibr 212 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  \/  x  =  y  \/  y R x ) )
2316, 17, 22issoi 4672 1  |-  R  Or  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965    e. wcel 1756   class class class wbr 4292    Or wor 4640
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-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2568  df-ral 2720  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-po 4641  df-so 4642
This theorem is referenced by:  ltsonq  9138  ltsosr  9261  ltso  9455  xrltso  11118
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