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Theorem isso2i 4449
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 eqid 2366 . . . . 5  |-  x  =  x
21orci 379 . . . 4  |-  ( x  =  x  \/  x R x )
3 eleq1 2426 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  A  <->  x  e.  A ) )
43anbi2d 684 . . . . . 6  |-  ( y  =  x  ->  (
( x  e.  A  /\  y  e.  A
)  <->  ( x  e.  A  /\  x  e.  A ) ) )
5 eqeq2 2375 . . . . . . . 8  |-  ( y  =  x  ->  (
x  =  y  <->  x  =  x ) )
6 breq1 4128 . . . . . . . 8  |-  ( y  =  x  ->  (
y R x  <->  x R x ) )
75, 6orbi12d 690 . . . . . . 7  |-  ( y  =  x  ->  (
( x  =  y  \/  y R x )  <->  ( x  =  x  \/  x R x ) ) )
8 breq2 4129 . . . . . . . 8  |-  ( y  =  x  ->  (
x R y  <->  x R x ) )
98notbid 285 . . . . . . 7  |-  ( y  =  x  ->  ( -.  x R y  <->  -.  x R x ) )
107, 9bibi12d 312 . . . . . 6  |-  ( y  =  x  ->  (
( ( x  =  y  \/  y R x )  <->  -.  x R y )  <->  ( (
x  =  x  \/  x R x )  <->  -.  x R x ) ) )
114, 10imbi12d 311 . . . . 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 319 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( ( x  =  y  \/  y R x )  <->  -.  x R y ) )
1411, 13chvarv 2026 . . . 4  |-  ( ( x  e.  A  /\  x  e.  A )  ->  ( ( x  =  x  \/  x R x )  <->  -.  x R x ) )
152, 14mpbii 202 . . 3  |-  ( ( x  e.  A  /\  x  e.  A )  ->  -.  x R x )
1615anidms 626 . 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 214 . . 3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( -.  x R y  ->  ( x  =  y  \/  y R x ) ) )
19 3orass 938 . . . 4  |-  ( ( x R y  \/  x  =  y  \/  y R x )  <-> 
( x R y  \/  ( x  =  y  \/  y R x ) ) )
20 df-or 359 . . . 4  |-  ( ( x R y  \/  ( x  =  y  \/  y R x ) )  <->  ( -.  x R y  ->  (
x  =  y  \/  y R x ) ) )
2119, 20bitri 240 . . 3  |-  ( ( x R y  \/  x  =  y  \/  y R x )  <-> 
( -.  x R y  ->  ( x  =  y  \/  y R x ) ) )
2218, 21sylibr 203 . 2  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  \/  x  =  y  \/  y R x ) )
2316, 17, 22issoi 4448 1  |-  R  Or  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 934    /\ w3a 935    = wceq 1647    e. wcel 1715   class class class wbr 4125    Or wor 4416
This theorem is referenced by:  ltsonq  8740  ltsosr  8863  ltso  9050  xrltso  10627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-po 4417  df-so 4418
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