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Theorem isso2i 4538
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 1689 . . . . 5  |-  x  =  x
21orci 381 . . . 4  |-  ( x  =  x  \/  x R x )
3 eleq1 2498 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  A  <->  x  e.  A ) )
43anbi2d 686 . . . . . 6  |-  ( y  =  x  ->  (
( x  e.  A  /\  y  e.  A
)  <->  ( x  e.  A  /\  x  e.  A ) ) )
5 equequ2 1699 . . . . . . . 8  |-  ( y  =  x  ->  (
x  =  y  <->  x  =  x ) )
6 breq1 4218 . . . . . . . 8  |-  ( y  =  x  ->  (
y R x  <->  x R x ) )
75, 6orbi12d 692 . . . . . . 7  |-  ( y  =  x  ->  (
( x  =  y  \/  y R x )  <->  ( x  =  x  \/  x R x ) ) )
8 breq2 4219 . . . . . . . 8  |-  ( y  =  x  ->  (
x R y  <->  x R x ) )
98notbid 287 . . . . . . 7  |-  ( y  =  x  ->  ( -.  x R y  <->  -.  x R x ) )
107, 9bibi12d 314 . . . . . 6  |-  ( y  =  x  ->  (
( ( x  =  y  \/  y R x )  <->  -.  x R y )  <->  ( (
x  =  x  \/  x R x )  <->  -.  x R x ) ) )
114, 10imbi12d 313 . . . . 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 321 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( x  =  y  \/  y R x )  <->  -.  x R y ) )
1411, 13chvarv 1970 . . . 4  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( ( x  =  x  \/  x R x )  <->  -.  x R x ) )
152, 14mpbii 204 . . 3  |-  ( ( x  e.  A  /\  x  e.  A )  ->  -.  x R x )
1615anidms 628 . 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 216 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( -.  x R y  ->  ( x  =  y  \/  y R x ) ) )
19 3orass 940 . . . 4  |-  ( ( x R y  \/  x  =  y  \/  y R x )  <-> 
( x R y  \/  ( x  =  y  \/  y R x ) ) )
20 df-or 361 . . . 4  |-  ( ( x R y  \/  ( x  =  y  \/  y R x ) )  <->  ( -.  x R y  ->  (
x  =  y  \/  y R x ) ) )
2119, 20bitri 242 . . 3  |-  ( ( x R y  \/  x  =  y  \/  y R x )  <-> 
( -.  x R y  ->  ( x  =  y  \/  y R x ) ) )
2218, 21sylibr 205 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  \/  x  =  y  \/  y R x ) )
2316, 17, 22issoi 4537 1  |-  R  Or  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    \/ w3o 936    /\ w3a 937    e. wcel 1726   class class class wbr 4215    Or wor 4505
This theorem is referenced by:  ltsonq  8851  ltsosr  8974  ltso  9161  xrltso  10739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-po 4506  df-so 4507
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