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Theorem onintonm 4516
Description: The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
onintonm  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  e.  On )
Distinct variable group:    x, A

Proof of Theorem onintonm
StepHypRef Expression
1 ssel 3149 . . . . . . 7  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  On ) )
2 eloni 4375 . . . . . . . 8  |-  ( x  e.  On  ->  Ord  x )
3 ordtr 4378 . . . . . . . 8  |-  ( Ord  x  ->  Tr  x
)
42, 3syl 14 . . . . . . 7  |-  ( x  e.  On  ->  Tr  x )
51, 4syl6 33 . . . . . 6  |-  ( A 
C_  On  ->  ( x  e.  A  ->  Tr  x ) )
65ralrimiv 2549 . . . . 5  |-  ( A 
C_  On  ->  A. x  e.  A  Tr  x
)
7 trint 4116 . . . . 5  |-  ( A. x  e.  A  Tr  x  ->  Tr  |^| A )
86, 7syl 14 . . . 4  |-  ( A 
C_  On  ->  Tr  |^| A )
98adantr 276 . . 3  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  Tr  |^| A
)
10 nfv 1528 . . . . 5  |-  F/ x  A  C_  On
11 nfe1 1496 . . . . 5  |-  F/ x E. x  x  e.  A
1210, 11nfan 1565 . . . 4  |-  F/ x
( A  C_  On  /\ 
E. x  x  e.  A )
13 intssuni2m 3868 . . . . . . . 8  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  C_  U. On )
14 unon 4510 . . . . . . . 8  |-  U. On  =  On
1513, 14sseqtrdi 3203 . . . . . . 7  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  C_  On )
1615sseld 3154 . . . . . 6  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  ( x  e.  |^| A  ->  x  e.  On ) )
1716, 2syl6 33 . . . . 5  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  ( x  e.  |^| A  ->  Ord  x ) )
1817, 3syl6 33 . . . 4  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  ( x  e.  |^| A  ->  Tr  x ) )
1912, 18ralrimi 2548 . . 3  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  A. x  e.  |^| A Tr  x
)
20 dford3 4367 . . 3  |-  ( Ord  |^| A  <->  ( Tr  |^| A  /\  A. x  e. 
|^| A Tr  x
) )
219, 19, 20sylanbrc 417 . 2  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  Ord  |^| A
)
22 inteximm 4149 . . . 4  |-  ( E. x  x  e.  A  ->  |^| A  e.  _V )
2322adantl 277 . . 3  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  e. 
_V )
24 elong 4373 . . 3  |-  ( |^| A  e.  _V  ->  (
|^| A  e.  On  <->  Ord  |^| A ) )
2523, 24syl 14 . 2  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  ( |^| A  e.  On  <->  Ord  |^| A
) )
2621, 25mpbird 167 1  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   E.wex 1492    e. wcel 2148   A.wral 2455   _Vcvv 2737    C_ wss 3129   U.cuni 3809   |^|cint 3844   Tr wtr 4101   Ord word 4362   Oncon0 4363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371
This theorem is referenced by:  onintrab2im  4517
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