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Theorem ordtri3or 4605
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 4603 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
2 ordirr 4591 . . . . . 6  |-  ( Ord  ( A  i^i  B
)  ->  -.  ( A  i^i  B )  e.  ( A  i^i  B
) )
31, 2syl 16 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  -.  ( A  i^i  B )  e.  ( A  i^i  B
) )
4 ianor 475 . . . . . 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 3522 . . . . . . 7  |-  ( ( A  i^i  B )  e.  ( A  i^i  B )  <->  ( ( A  i^i  B )  e.  A  /\  ( A  i^i  B )  e.  B ) )
6 incom 3525 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
76eleq1i 2498 . . . . . . . 8  |-  ( ( A  i^i  B )  e.  B  <->  ( B  i^i  A )  e.  B
)
87anbi2i 676 . . . . . . 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 241 . . . . . 6  |-  ( ( A  i^i  B )  e.  ( A  i^i  B )  <->  ( ( A  i^i  B )  e.  A  /\  ( B  i^i  A )  e.  B ) )
104, 9xchnxbir 301 . . . . 5  |-  ( -.  ( A  i^i  B
)  e.  ( A  i^i  B )  <->  ( -.  ( A  i^i  B )  e.  A  \/  -.  ( B  i^i  A )  e.  B ) )
113, 10sylib 189 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( A  i^i  B )  e.  A  \/  -.  ( B  i^i  A )  e.  B ) )
12 inss1 3553 . . . . . . . . . 10  |-  ( A  i^i  B )  C_  A
13 ordsseleq 4602 . . . . . . . . . 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 203 . . . . . . . . 9  |-  ( ( Ord  ( A  i^i  B )  /\  Ord  A
)  ->  ( ( A  i^i  B )  e.  A  \/  ( A  i^i  B )  =  A ) )
151, 14sylan 458 . . . . . . . 8  |-  ( ( ( Ord  A  /\  Ord  B )  /\  Ord  A )  ->  ( ( A  i^i  B )  e.  A  \/  ( A  i^i  B )  =  A ) )
1615anabss1 788 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  i^i  B )  e.  A  \/  ( A  i^i  B )  =  A ) )
1716ord 367 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( A  i^i  B )  e.  A  ->  ( A  i^i  B )  =  A ) )
18 df-ss 3326 . . . . . 6  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
1917, 18syl6ibr 219 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( A  i^i  B )  e.  A  ->  A  C_  B ) )
20 ordin 4603 . . . . . . . . 9  |-  ( ( Ord  B  /\  Ord  A )  ->  Ord  ( B  i^i  A ) )
21 inss1 3553 . . . . . . . . . 10  |-  ( B  i^i  A )  C_  B
22 ordsseleq 4602 . . . . . . . . . 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 203 . . . . . . . . 9  |-  ( ( Ord  ( B  i^i  A )  /\  Ord  B
)  ->  ( ( B  i^i  A )  e.  B  \/  ( B  i^i  A )  =  B ) )
2420, 23sylan 458 . . . . . . . 8  |-  ( ( ( Ord  B  /\  Ord  A )  /\  Ord  B )  ->  ( ( B  i^i  A )  e.  B  \/  ( B  i^i  A )  =  B ) )
2524anabss4 789 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( B  i^i  A )  e.  B  \/  ( B  i^i  A )  =  B ) )
2625ord 367 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( B  i^i  A )  e.  B  ->  ( B  i^i  A )  =  B ) )
27 df-ss 3326 . . . . . 6  |-  ( B 
C_  A  <->  ( B  i^i  A )  =  B )
2826, 27syl6ibr 219 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( -.  ( B  i^i  A )  e.  B  ->  B  C_  A ) )
2919, 28orim12d 812 . . . 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 15 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C_  B  \/  B  C_  A ) )
31 sspsstri 3441 . . 3  |-  ( ( A  C_  B  \/  B  C_  A )  <->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
3230, 31sylib 189 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) )
33 ordelpss 4601 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  A  C.  B ) )
34 biidd 229 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  B  <->  A  =  B
) )
35 ordelpss 4601 . . . 4  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  e.  A  <->  B  C.  A ) )
3635ancoms 440 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( B  e.  A  <->  B  C.  A ) )
3733, 34, 363orbi123d 1253 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  e.  B  \/  A  =  B  \/  B  e.  A )  <->  ( A  C.  B  \/  A  =  B  \/  B  C.  A ) ) )
3832, 37mpbird 224 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 177    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312    C. wpss 3313   Ord word 4572
This theorem is referenced by:  ordtri1  4606  ordtri3  4609  ordon  4755  ordeleqon  4761  smo11  6618  smoord  6619  omopth2  6819  r111  7693  tcrank  7800  domtriomlem  8314  axdc3lem2  8323  zorn2lem6  8373  grur1  8687  poseq  25520  soseq  25521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576
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