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Theorem oninton 6620
Description: The intersection of a nonempty 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 6615 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
21ex 434 . . 3  |-  ( A 
C_  On  ->  ( A  =/=  (/)  ->  |^| A  e.  A ) )
3 ssel 3483 . . 3  |-  ( A 
C_  On  ->  ( |^| A  e.  A  ->  |^| A  e.  On ) )
42, 3syld 44 . 2  |-  ( A 
C_  On  ->  ( A  =/=  (/)  ->  |^| A  e.  On ) )
54imp 429 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1804    =/= wne 2638    C_ wss 3461   (/)c0 3770   |^|cint 4271   Oncon0 4868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872
This theorem is referenced by:  onintrab  6621  onnmin  6623  onminex  6627  onmindif2  6632  iinon  7013  oawordeulem  7205  nnawordex  7288  tz9.12lem1  8208  rankf  8215  cardf2  8327  cff  8631  coftr  8656  sltval2  29391  nodenselem4  29419  nocvxminlem  29425  dnnumch3lem  30967  dnnumch3  30968
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