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Theorem ordtri3or 4426
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 4424 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
2 ordirr 4412 . . . . . 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 3360 . . . . . . 7  |-  ( ( A  i^i  B )  e.  ( A  i^i  B )  <->  ( ( A  i^i  B )  e.  A  /\  ( A  i^i  B )  e.  B ) )
6 incom 3363 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
76eleq1i 2348 . . . . . . . 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 3391 . . . . . . . . . 10  |-  ( A  i^i  B )  C_  A
13 ordsseleq 4423 . . . . . . . . . 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 3168 . . . . . 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 4424 . . . . . . . . 9  |-  ( ( Ord  B  /\  Ord  A )  ->  Ord  ( B  i^i  A ) )
21 inss1 3391 . . . . . . . . . 10  |-  ( B  i^i  A )  C_  B
22 ordsseleq 4423 . . . . . . . . . 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 3168 . . . . . 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 3280 . . 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 4422 . . 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 4422 . . . 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 1625    e. wcel 1686    i^i cin 3153    C_ wss 3154    C. wpss 3155   Ord word 4393
This theorem is referenced by:  ordtri1  4427  ordtri3  4430  ordon  4576  ordeleqon  4582  smo11  6383  smoord  6384  omopth2  6584  r111  7449  tcrank  7556  domtriomlem  8070  axdc3lem2  8079  zorn2lem6  8130  grur1  8444  poseq  24255  soseq  24256  celsor  25122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4307  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397
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