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Theorem onintonm 4306
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 3008 . . . . . . 7  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  On ) )
2 eloni 4175 . . . . . . . 8  |-  ( x  e.  On  ->  Ord  x )
3 ordtr 4178 . . . . . . . 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 2441 . . . . 5  |-  ( A 
C_  On  ->  A. x  e.  A  Tr  x
)
7 trint 3925 . . . . 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 1464 . . . . 5  |-  F/ x  A  C_  On
11 nfe1 1428 . . . . 5  |-  F/ x E. x  x  e.  A
1210, 11nfan 1500 . . . 4  |-  F/ x
( A  C_  On  /\ 
E. x  x  e.  A )
13 intssuni2m 3695 . . . . . . . 8  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  C_  U. On )
14 unon 4300 . . . . . . . 8  |-  U. On  =  On
1513, 14syl6sseq 3061 . . . . . . 7  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  C_  On )
1615sseld 3013 . . . . . 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 2440 . . 3  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  A. x  e.  |^| A Tr  x
)
20 dford3 4167 . . 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 3959 . . . 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 4173 . . 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 1424    e. wcel 1436   A.wral 2355   _Vcvv 2615    C_ wss 2988   U.cuni 3636   |^|cint 3671   Tr wtr 3910   Ord word 4162   Oncon0 4163
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-pow 3983  ax-pr 4009  ax-un 4233
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-uni 3637  df-int 3672  df-tr 3911  df-iord 4166  df-on 4168  df-suc 4171
This theorem is referenced by:  onintrab2im  4307
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