MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oninton Unicode version

Theorem oninton 4771
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 4766 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  A )
21ex 424 . . 3  |-  ( A 
C_  On  ->  ( A  =/=  (/)  ->  |^| A  e.  A ) )
3 ssel 3334 . . 3  |-  ( A 
C_  On  ->  ( |^| A  e.  A  ->  |^| A  e.  On ) )
42, 3syld 42 . 2  |-  ( A 
C_  On  ->  ( A  =/=  (/)  ->  |^| A  e.  On ) )
54imp 419 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    =/= wne 2598    C_ wss 3312   (/)c0 3620   |^|cint 4042   Oncon0 4573
This theorem is referenced by:  onintrab  4772  onnmin  4774  onminex  4778  onmindif2  4783  iinon  6593  oawordeulem  6788  nnawordex  6871  tz9.12lem1  7702  rankf  7709  cardf2  7819  cff  8117  coftr  8142  sltval2  25565  nodenselem4  25593  nocvxminlem  25599  dnnumch3lem  27058  dnnumch3  27059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
  Copyright terms: Public domain W3C validator