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Theorem onintonm 4269
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 2994 . . . . . . 7  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  On ) )
2 eloni 4138 . . . . . . . 8  |-  ( x  e.  On  ->  Ord  x )
3 ordtr 4141 . . . . . . . 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 2434 . . . . 5  |-  ( A 
C_  On  ->  A. x  e.  A  Tr  x
)
7 trint 3898 . . . . 5  |-  ( A. x  e.  A  Tr  x  ->  Tr  |^| A )
86, 7syl 14 . . . 4  |-  ( A 
C_  On  ->  Tr  |^| A )
98adantr 270 . . 3  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  Tr  |^| A
)
10 nfv 1462 . . . . 5  |-  F/ x  A  C_  On
11 nfe1 1426 . . . . 5  |-  F/ x E. x  x  e.  A
1210, 11nfan 1498 . . . 4  |-  F/ x
( A  C_  On  /\ 
E. x  x  e.  A )
13 intssuni2m 3668 . . . . . . . 8  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  C_  U. On )
14 unon 4263 . . . . . . . 8  |-  U. On  =  On
1513, 14syl6sseq 3046 . . . . . . 7  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  C_  On )
1615sseld 2999 . . . . . 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 2433 . . 3  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  A. x  e.  |^| A Tr  x
)
20 dford3 4130 . . 3  |-  ( Ord  |^| A  <->  ( Tr  |^| A  /\  A. x  e. 
|^| A Tr  x
) )
219, 19, 20sylanbrc 408 . 2  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  Ord  |^| A
)
22 inteximm 3932 . . . 4  |-  ( E. x  x  e.  A  ->  |^| A  e.  _V )
2322adantl 271 . . 3  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  e. 
_V )
24 elong 4136 . . 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 165 1  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   E.wex 1422    e. wcel 1434   A.wral 2349   _Vcvv 2602    C_ wss 2974   U.cuni 3609   |^|cint 3644   Tr wtr 3883   Ord word 4125   Oncon0 4126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-uni 3610  df-int 3645  df-tr 3884  df-iord 4129  df-on 4131  df-suc 4134
This theorem is referenced by:  onintrab2im  4270
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