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Theorem ordtri3or 4424
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 4422 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
2 ordirr 4410 . . . . . 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 3358 . . . . . . 7  |-  ( ( A  i^i  B )  e.  ( A  i^i  B )  <->  ( ( A  i^i  B )  e.  A  /\  ( A  i^i  B )  e.  B ) )
6 incom 3361 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
76eleq1i 2346 . . . . . . . 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 3389 . . . . . . . . . 10  |-  ( A  i^i  B )  C_  A
13 ordsseleq 4421 . . . . . . . . . 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 3166 . . . . . 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 4422 . . . . . . . . 9  |-  ( ( Ord  B  /\  Ord  A )  ->  Ord  ( B  i^i  A ) )
21 inss1 3389 . . . . . . . . . 10  |-  ( B  i^i  A )  C_  B
22 ordsseleq 4421 . . . . . . . . . 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 3166 . . . . . 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 3278 . . 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 4420 . . 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 4420 . . . 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 1251 . 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 933    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152    C. wpss 3153   Ord word 4391
This theorem is referenced by:  ordtri1  4425  ordtri3  4428  ordon  4574  ordeleqon  4580  smo11  6381  smoord  6382  omopth2  6582  r111  7447  tcrank  7554  domtriomlem  8068  axdc3lem2  8077  zorn2lem6  8128  grur1  8442  poseq  24253  soseq  24254  celsor  25111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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