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Theorem ordelinel 4493
Description: The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
ordelinel  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( A  i^i  B )  e.  C  <->  ( A  e.  C  \/  B  e.  C ) ) )

Proof of Theorem ordelinel
StepHypRef Expression
1 ordtri2or3 4492 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  ( A  i^i  B )  \/  B  =  ( A  i^i  B
) ) )
213adant3 975 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( A  =  ( A  i^i  B
)  \/  B  =  ( A  i^i  B
) ) )
3 eleq1 2345 . . . . 5  |-  ( A  =  ( A  i^i  B )  ->  ( A  e.  C  <->  ( A  i^i  B )  e.  C ) )
4 orc 374 . . . . 5  |-  ( A  e.  C  ->  ( A  e.  C  \/  B  e.  C )
)
53, 4syl6bir 220 . . . 4  |-  ( A  =  ( A  i^i  B )  ->  ( ( A  i^i  B )  e.  C  ->  ( A  e.  C  \/  B  e.  C ) ) )
6 eleq1 2345 . . . . 5  |-  ( B  =  ( A  i^i  B )  ->  ( B  e.  C  <->  ( A  i^i  B )  e.  C ) )
7 olc 373 . . . . 5  |-  ( B  e.  C  ->  ( A  e.  C  \/  B  e.  C )
)
86, 7syl6bir 220 . . . 4  |-  ( B  =  ( A  i^i  B )  ->  ( ( A  i^i  B )  e.  C  ->  ( A  e.  C  \/  B  e.  C ) ) )
95, 8jaoi 368 . . 3  |-  ( ( A  =  ( A  i^i  B )  \/  B  =  ( A  i^i  B ) )  ->  ( ( A  i^i  B )  e.  C  ->  ( A  e.  C  \/  B  e.  C ) ) )
102, 9syl 15 . 2  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( A  i^i  B )  e.  C  ->  ( A  e.  C  \/  B  e.  C ) ) )
11 inss1 3391 . . . 4  |-  ( A  i^i  B )  C_  A
12 ordin 4424 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
1312anim1i 551 . . . . . 6  |-  ( ( ( Ord  A  /\  Ord  B )  /\  Ord  C )  ->  ( Ord  ( A  i^i  B )  /\  Ord  C ) )
14133impa 1146 . . . . 5  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( Ord  ( A  i^i  B )  /\  Ord  C ) )
15 ordtr2 4438 . . . . 5  |-  ( ( Ord  ( A  i^i  B )  /\  Ord  C
)  ->  ( (
( A  i^i  B
)  C_  A  /\  A  e.  C )  ->  ( A  i^i  B
)  e.  C ) )
1614, 15syl 15 . . . 4  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( ( A  i^i  B ) 
C_  A  /\  A  e.  C )  ->  ( A  i^i  B )  e.  C ) )
1711, 16mpani 657 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( A  e.  C  ->  ( A  i^i  B )  e.  C
) )
18 inss2 3392 . . . 4  |-  ( A  i^i  B )  C_  B
19 ordtr2 4438 . . . . 5  |-  ( ( Ord  ( A  i^i  B )  /\  Ord  C
)  ->  ( (
( A  i^i  B
)  C_  B  /\  B  e.  C )  ->  ( A  i^i  B
)  e.  C ) )
2014, 19syl 15 . . . 4  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( ( A  i^i  B ) 
C_  B  /\  B  e.  C )  ->  ( A  i^i  B )  e.  C ) )
2118, 20mpani 657 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( B  e.  C  ->  ( A  i^i  B )  e.  C
) )
2217, 21jaod 369 . 2  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( A  e.  C  \/  B  e.  C )  ->  ( A  i^i  B )  e.  C ) )
2310, 22impbid 183 1  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( A  i^i  B )  e.  C  <->  ( A  e.  C  \/  B  e.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    i^i cin 3153    C_ wss 3154   Ord word 4393
This theorem is referenced by:  mreexexd  13552
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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