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Theorem onintonm 4403
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 3061 . . . . . . 7  |-  ( A 
C_  On  ->  ( x  e.  A  ->  x  e.  On ) )
2 eloni 4267 . . . . . . . 8  |-  ( x  e.  On  ->  Ord  x )
3 ordtr 4270 . . . . . . . 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 2481 . . . . 5  |-  ( A 
C_  On  ->  A. x  e.  A  Tr  x
)
7 trint 4011 . . . . 5  |-  ( A. x  e.  A  Tr  x  ->  Tr  |^| A )
86, 7syl 14 . . . 4  |-  ( A 
C_  On  ->  Tr  |^| A )
98adantr 274 . . 3  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  Tr  |^| A
)
10 nfv 1493 . . . . 5  |-  F/ x  A  C_  On
11 nfe1 1457 . . . . 5  |-  F/ x E. x  x  e.  A
1210, 11nfan 1529 . . . 4  |-  F/ x
( A  C_  On  /\ 
E. x  x  e.  A )
13 intssuni2m 3765 . . . . . . . 8  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  C_  U. On )
14 unon 4397 . . . . . . . 8  |-  U. On  =  On
1513, 14sseqtrdi 3115 . . . . . . 7  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  C_  On )
1615sseld 3066 . . . . . 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 2480 . . 3  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  A. x  e.  |^| A Tr  x
)
20 dford3 4259 . . 3  |-  ( Ord  |^| A  <->  ( Tr  |^| A  /\  A. x  e. 
|^| A Tr  x
) )
219, 19, 20sylanbrc 413 . 2  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  Ord  |^| A
)
22 inteximm 4044 . . . 4  |-  ( E. x  x  e.  A  ->  |^| A  e.  _V )
2322adantl 275 . . 3  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  e. 
_V )
24 elong 4265 . . 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 166 1  |-  ( ( A  C_  On  /\  E. x  x  e.  A
)  ->  |^| A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1453    e. wcel 1465   A.wral 2393   _Vcvv 2660    C_ wss 3041   U.cuni 3706   |^|cint 3741   Tr wtr 3996   Ord word 4254   Oncon0 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263
This theorem is referenced by:  onintrab2im  4404
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