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Theorem ordelinel 4680
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 4679 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  ( A  i^i  B )  \/  B  =  ( A  i^i  B
) ) )
213adant3 977 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( A  =  ( A  i^i  B
)  \/  B  =  ( A  i^i  B
) ) )
3 eleq1 2496 . . . . 5  |-  ( A  =  ( A  i^i  B )  ->  ( A  e.  C  <->  ( A  i^i  B )  e.  C ) )
4 orc 375 . . . . 5  |-  ( A  e.  C  ->  ( A  e.  C  \/  B  e.  C )
)
53, 4syl6bir 221 . . . 4  |-  ( A  =  ( A  i^i  B )  ->  ( ( A  i^i  B )  e.  C  ->  ( A  e.  C  \/  B  e.  C ) ) )
6 eleq1 2496 . . . . 5  |-  ( B  =  ( A  i^i  B )  ->  ( B  e.  C  <->  ( A  i^i  B )  e.  C ) )
7 olc 374 . . . . 5  |-  ( B  e.  C  ->  ( A  e.  C  \/  B  e.  C )
)
86, 7syl6bir 221 . . . 4  |-  ( B  =  ( A  i^i  B )  ->  ( ( A  i^i  B )  e.  C  ->  ( A  e.  C  \/  B  e.  C ) ) )
95, 8jaoi 369 . . 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 16 . 2  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( A  i^i  B )  e.  C  ->  ( A  e.  C  \/  B  e.  C ) ) )
11 inss1 3561 . . . 4  |-  ( A  i^i  B )  C_  A
12 ordin 4611 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
1312anim1i 552 . . . . . 6  |-  ( ( ( Ord  A  /\  Ord  B )  /\  Ord  C )  ->  ( Ord  ( A  i^i  B )  /\  Ord  C ) )
14133impa 1148 . . . . 5  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( Ord  ( A  i^i  B )  /\  Ord  C ) )
15 ordtr2 4625 . . . . 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 16 . . . 4  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( ( A  i^i  B ) 
C_  A  /\  A  e.  C )  ->  ( A  i^i  B )  e.  C ) )
1711, 16mpani 658 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( A  e.  C  ->  ( A  i^i  B )  e.  C
) )
18 inss2 3562 . . . 4  |-  ( A  i^i  B )  C_  B
19 ordtr2 4625 . . . . 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 16 . . . 4  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( ( A  i^i  B ) 
C_  B  /\  B  e.  C )  ->  ( A  i^i  B )  e.  C ) )
2118, 20mpani 658 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( B  e.  C  ->  ( A  i^i  B )  e.  C
) )
2217, 21jaod 370 . 2  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( A  e.  C  \/  B  e.  C )  ->  ( A  i^i  B )  e.  C ) )
2310, 22impbid 184 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   Ord word 4580
This theorem is referenced by:  mreexexd  13873
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584
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