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Theorem oninton 6511
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 6506 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
21ex 434 . . 3  |-  ( A 
C_  On  ->  ( A  =/=  (/)  ->  |^| A  e.  A ) )
3 ssel 3448 . . 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 1758    =/= wne 2644    C_ wss 3426   (/)c0 3735   |^|cint 4226   Oncon0 4817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-br 4391  df-opab 4449  df-tr 4484  df-eprel 4730  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821
This theorem is referenced by:  onintrab  6512  onnmin  6514  onminex  6518  onmindif2  6523  iinon  6901  oawordeulem  7093  nnawordex  7176  tz9.12lem1  8095  rankf  8102  cardf2  8214  cff  8518  coftr  8543  sltval2  27931  nodenselem4  27959  nocvxminlem  27965  dnnumch3lem  29537  dnnumch3  29538
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