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Theorem ordtri3or 4547
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 4545 . . . . . 6  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
2 ordirr 4533 . . . . . 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 3466 . . . . . . 7  |-  ( ( A  i^i  B )  e.  ( A  i^i  B )  <->  ( ( A  i^i  B )  e.  A  /\  ( A  i^i  B )  e.  B ) )
6 incom 3469 . . . . . . . . 9  |-  ( A  i^i  B )  =  ( B  i^i  A
)
76eleq1i 2443 . . . . . . . 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 3497 . . . . . . . . . 10  |-  ( A  i^i  B )  C_  A
13 ordsseleq 4544 . . . . . . . . . 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 3270 . . . . . 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 4545 . . . . . . . . 9  |-  ( ( Ord  B  /\  Ord  A )  ->  Ord  ( B  i^i  A ) )
21 inss1 3497 . . . . . . . . . 10  |-  ( B  i^i  A )  C_  B
22 ordsseleq 4544 . . . . . . . . . 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 3270 . . . . . 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 3385 . . 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 4543 . . 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 4543 . . . 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 1649    e. wcel 1717    i^i cin 3255    C_ wss 3256    C. wpss 3257   Ord word 4514
This theorem is referenced by:  ordtri1  4548  ordtri3  4551  ordon  4696  ordeleqon  4702  smo11  6555  smoord  6556  omopth2  6756  r111  7627  tcrank  7734  domtriomlem  8248  axdc3lem2  8257  zorn2lem6  8307  grur1  8621  poseq  25270  soseq  25271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518
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