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Theorem isso2i 4807
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 1842 . . . . 5  |-  x  =  x
21orci 391 . . . 4  |-  ( x  =  x  \/  x R x )
3 eleq1 2501 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  A  <->  x  e.  A ) )
43anbi2d 708 . . . . . 6  |-  ( y  =  x  ->  (
( x  e.  A  /\  y  e.  A
)  <->  ( x  e.  A  /\  x  e.  A ) ) )
5 equequ2 1851 . . . . . . . 8  |-  ( y  =  x  ->  (
x  =  y  <->  x  =  x ) )
6 breq1 4429 . . . . . . . 8  |-  ( y  =  x  ->  (
y R x  <->  x R x ) )
75, 6orbi12d 714 . . . . . . 7  |-  ( y  =  x  ->  (
( x  =  y  \/  y R x )  <->  ( x  =  x  \/  x R x ) ) )
8 breq2 4430 . . . . . . . 8  |-  ( y  =  x  ->  (
x R y  <->  x R x ) )
98notbid 295 . . . . . . 7  |-  ( y  =  x  ->  ( -.  x R y  <->  -.  x R x ) )
107, 9bibi12d 322 . . . . . 6  |-  ( y  =  x  ->  (
( ( x  =  y  \/  y R x )  <->  -.  x R y )  <->  ( (
x  =  x  \/  x R x )  <->  -.  x R x ) ) )
114, 10imbi12d 321 . . . . 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 330 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( x  =  y  \/  y R x )  <->  -.  x R y ) )
1411, 13chvarv 2070 . . . 4  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( ( x  =  x  \/  x R x )  <->  -.  x R x ) )
152, 14mpbii 214 . . 3  |-  ( ( x  e.  A  /\  x  e.  A )  ->  -.  x R x )
1615anidms 649 . 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 226 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( -.  x R y  ->  ( x  =  y  \/  y R x ) ) )
19 3orass 985 . . . 4  |-  ( ( x R y  \/  x  =  y  \/  y R x )  <-> 
( x R y  \/  ( x  =  y  \/  y R x ) ) )
20 df-or 371 . . . 4  |-  ( ( x R y  \/  ( x  =  y  \/  y R x ) )  <->  ( -.  x R y  ->  (
x  =  y  \/  y R x ) ) )
2119, 20bitri 252 . . 3  |-  ( ( x R y  \/  x  =  y  \/  y R x )  <-> 
( -.  x R y  ->  ( x  =  y  \/  y R x ) ) )
2218, 21sylibr 215 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  \/  x  =  y  \/  y R x ) )
2316, 17, 22issoi 4806 1  |-  R  Or  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981    /\ w3a 982    e. wcel 1870   class class class wbr 4426    Or wor 4774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-po 4775  df-so 4776
This theorem is referenced by:  ltsonq  9393  ltsosr  9517  ltso  9713  xrltso  11440
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