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Theorem oninton 4591
Description: The intersection of a non-empty collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by NM, 29-Jan-1997.)
Assertion
Ref Expression
oninton  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )

Proof of Theorem oninton
StepHypRef Expression
1 onint 4586 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
21ex 423 . . 3  |-  ( A 
C_  On  ->  ( A  =/=  (/)  ->  |^| A  e.  A ) )
3 ssel 3174 . . 3  |-  ( A 
C_  On  ->  ( |^| A  e.  A  ->  |^| A  e.  On ) )
42, 3syld 40 . 2  |-  ( A 
C_  On  ->  ( A  =/=  (/)  ->  |^| A  e.  On ) )
54imp 418 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   |^|cint 3862   Oncon0 4392
This theorem is referenced by:  onintrab  4592  onnmin  4594  onminex  4598  onmindif2  4603  iinon  6357  oawordeulem  6552  nnawordex  6635  tz9.12lem1  7459  rankf  7466  cardf2  7576  cff  7874  coftr  7899  sltval2  24310  nodenselem4  24338  nocvxminlem  24344  dnnumch3lem  27143  dnnumch3  27144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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