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Theorem ordtri3or 4527
Description: A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. (Contributed by NM, 10-May-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtri3or  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )

Proof of Theorem ordtri3or
StepHypRef Expression
1 ordin 4525 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
2 ordirr 4513 . . . . . 6  |-  ( Ord  ( A  i^i  B
)  ->  -.  ( A  i^i  B )  e.  ( A  i^i  B
) )
31, 2syl 15 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  -.  ( A  i^i  B )  e.  ( A  i^i  B
) )
4 ianor 474 . . . . . 6  |-  ( -.  ( ( A  i^i  B )  e.  A  /\  ( B  i^i  A )  e.  B )  <->  ( -.  ( A  i^i  B )  e.  A  \/  -.  ( B  i^i  A )  e.  B ) )
5 elin 3446 . . . . . . 7  |-  ( ( A  i^i  B )  e.  ( A  i^i  B )  <->  ( ( A  i^i  B )  e.  A  /\  ( A  i^i  B )  e.  B ) )
6 incom 3449 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
76eleq1i 2429 . . . . . . . 8  |-  ( ( A  i^i  B )  e.  B  <->  ( B  i^i  A )  e.  B
)
87anbi2i 675 . . . . . . 7  |-  ( ( ( A  i^i  B
)  e.  A  /\  ( A  i^i  B )  e.  B )  <->  ( ( A  i^i  B )  e.  A  /\  ( B  i^i  A )  e.  B ) )
95, 8bitri 240 . . . . . 6  |-  ( ( A  i^i  B )  e.  ( A  i^i  B )  <->  ( ( A  i^i  B )  e.  A  /\  ( B  i^i  A )  e.  B ) )
104, 9xchnxbir 300 . . . . 5  |-  ( -.  ( A  i^i  B
)  e.  ( A  i^i  B )  <->  ( -.  ( A  i^i  B )  e.  A  \/  -.  ( B  i^i  A )  e.  B ) )
113, 10sylib 188 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( A  i^i  B )  e.  A  \/  -.  ( B  i^i  A )  e.  B ) )
12 inss1 3477 . . . . . . . . . 10  |-  ( A  i^i  B )  C_  A
13 ordsseleq 4524 . . . . . . . . . 10  |-  ( ( Ord  ( A  i^i  B )  /\  Ord  A
)  ->  ( ( A  i^i  B )  C_  A 
<->  ( ( A  i^i  B )  e.  A  \/  ( A  i^i  B )  =  A ) ) )
1412, 13mpbii 202 . . . . . . . . 9  |-  ( ( Ord  ( A  i^i  B )  /\  Ord  A
)  ->  ( ( A  i^i  B )  e.  A  \/  ( A  i^i  B )  =  A ) )
151, 14sylan 457 . . . . . . . 8  |-  ( ( ( Ord  A  /\  Ord  B )  /\  Ord  A )  ->  ( ( A  i^i  B )  e.  A  \/  ( A  i^i  B )  =  A ) )
1615anabss1 787 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  i^i  B )  e.  A  \/  ( A  i^i  B )  =  A ) )
1716ord 366 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( A  i^i  B )  e.  A  ->  ( A  i^i  B )  =  A ) )
18 df-ss 3252 . . . . . 6  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
1917, 18syl6ibr 218 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( A  i^i  B )  e.  A  ->  A  C_  B ) )
20 ordin 4525 . . . . . . . . 9  |-  ( ( Ord  B  /\  Ord  A )  ->  Ord  ( B  i^i  A ) )
21 inss1 3477 . . . . . . . . . 10  |-  ( B  i^i  A )  C_  B
22 ordsseleq 4524 . . . . . . . . . 10  |-  ( ( Ord  ( B  i^i  A )  /\  Ord  B
)  ->  ( ( B  i^i  A )  C_  B 
<->  ( ( B  i^i  A )  e.  B  \/  ( B  i^i  A )  =  B ) ) )
2321, 22mpbii 202 . . . . . . . . 9  |-  ( ( Ord  ( B  i^i  A )  /\  Ord  B
)  ->  ( ( B  i^i  A )  e.  B  \/  ( B  i^i  A )  =  B ) )
2420, 23sylan 457 . . . . . . . 8  |-  ( ( ( Ord  B  /\  Ord  A )  /\  Ord  B )  ->  ( ( B  i^i  A )  e.  B  \/  ( B  i^i  A )  =  B ) )
2524anabss4 788 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( B  i^i  A )  e.  B  \/  ( B  i^i  A )  =  B ) )
2625ord 366 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( B  i^i  A )  e.  B  ->  ( B  i^i  A )  =  B ) )
27 df-ss 3252 . . . . . 6  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
2826, 27syl6ibr 218 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( B  i^i  A )  e.  B  ->  B  C_  A ) )
2919, 28orim12d 811 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( -.  ( A  i^i  B
)  e.  A  \/  -.  ( B  i^i  A
)  e.  B )  ->  ( A  C_  B  \/  B  C_  A
) ) )
3011, 29mpd 14 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  \/  B  C_  A ) )
31 sspsstri 3365 . . 3  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
3230, 31sylib 188 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
33 ordelpss 4523 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  A  C.  B ) )
34 biidd 228 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  A  =  B
) )
35 ordelpss 4523 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  e.  A  <->  B  C.  A ) )
3635ancoms 439 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  <->  B  C.  A ) )
3733, 34, 363orbi123d 1252 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B  \/  B  e.  A )  <->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) ) )
3832, 37mpbird 223 1  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  \/  A  =  B  \/  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 934    = wceq 1647    e. wcel 1715    i^i cin 3237    C_ wss 3238    C. wpss 3239   Ord word 4494
This theorem is referenced by:  ordtri1  4528  ordtri3  4531  ordon  4677  ordeleqon  4683  smo11  6523  smoord  6524  omopth2  6724  r111  7594  tcrank  7701  domtriomlem  8215  axdc3lem2  8224  zorn2lem6  8275  grur1  8589  poseq  24994  soseq  24995
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-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
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-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498
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