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Theorem ordelinel 4976
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 4975 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  =  ( A  i^i  B )  \/  B  =  ( A  i^i  B
) ) )
213adant3 1016 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( A  =  ( A  i^i  B
)  \/  B  =  ( A  i^i  B
) ) )
3 eleq1 2539 . . . . 5  |-  ( A  =  ( A  i^i  B )  ->  ( A  e.  C  <->  ( A  i^i  B )  e.  C ) )
4 orc 385 . . . . 5  |-  ( A  e.  C  ->  ( A  e.  C  \/  B  e.  C )
)
53, 4syl6bir 229 . . . 4  |-  ( A  =  ( A  i^i  B )  ->  ( ( A  i^i  B )  e.  C  ->  ( A  e.  C  \/  B  e.  C ) ) )
6 eleq1 2539 . . . . 5  |-  ( B  =  ( A  i^i  B )  ->  ( B  e.  C  <->  ( A  i^i  B )  e.  C ) )
7 olc 384 . . . . 5  |-  ( B  e.  C  ->  ( A  e.  C  \/  B  e.  C )
)
86, 7syl6bir 229 . . . 4  |-  ( B  =  ( A  i^i  B )  ->  ( ( A  i^i  B )  e.  C  ->  ( A  e.  C  \/  B  e.  C ) ) )
95, 8jaoi 379 . . 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 3718 . . . 4  |-  ( A  i^i  B )  C_  A
12 ordin 4908 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  i^i  B ) )
1312anim1i 568 . . . . . 6  |-  ( ( ( Ord  A  /\  Ord  B )  /\  Ord  C )  ->  ( Ord  ( A  i^i  B )  /\  Ord  C ) )
14133impa 1191 . . . . 5  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( Ord  ( A  i^i  B )  /\  Ord  C ) )
15 ordtr2 4922 . . . . 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 676 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( A  e.  C  ->  ( A  i^i  B )  e.  C
) )
18 inss2 3719 . . . 4  |-  ( A  i^i  B )  C_  B
19 ordtr2 4922 . . . . 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 676 . . 3  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( B  e.  C  ->  ( A  i^i  B )  e.  C
) )
2217, 21jaod 380 . 2  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( A  e.  C  \/  B  e.  C )  ->  ( A  i^i  B )  e.  C ) )
2310, 22impbid 191 1  |-  ( ( Ord  A  /\  Ord  B  /\  Ord  C )  ->  ( ( A  i^i  B )  e.  C  <->  ( A  e.  C  \/  B  e.  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   Ord word 4877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881
This theorem is referenced by:  mreexexd  14903
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