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Theorem oninton 4590
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 4585 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
21ex 425 . . 3  |-  ( A 
C_  On  ->  ( A  =/=  (/)  ->  |^| A  e.  A ) )
3 ssel 3175 . . 3  |-  ( A 
C_  On  ->  ( |^| A  e.  A  ->  |^| A  e.  On ) )
42, 3syld 42 . 2  |-  ( A 
C_  On  ->  ( A  =/=  (/)  ->  |^| A  e.  On ) )
54imp 420 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1685    =/= wne 2447    C_ wss 3153   (/)c0 3456   |^|cint 3863   Oncon0 4391
This theorem is referenced by:  onintrab  4591  onnmin  4593  onminex  4597  onmindif2  4602  iinon  6352  oawordeulem  6547  nnawordex  6630  tz9.12lem1  7454  rankf  7461  cardf2  7571  cff  7869  coftr  7894  sltval2  23710  axdenselem4  23739  nocvxminlem  23745  dnnumch3lem  26542  dnnumch3  26543
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395
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