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Theorem onss 6604
Description: An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
onss  |-  ( A  e.  On  ->  A  C_  On )

Proof of Theorem onss
StepHypRef Expression
1 eloni 4888 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordsson 6603 . 2  |-  ( Ord 
A  ->  A  C_  On )
31, 2syl 16 1  |-  ( A  e.  On  ->  A  C_  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767    C_ wss 3476   Ord word 4877   Oncon0 4878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882
This theorem is referenced by:  onuni  6606  onminex  6620  suceloni  6626  onssi  6650  tfi  6666  tfr3  7065  tz7.49  7107  tz7.49c  7108  oacomf1olem  7210  oeeulem  7247  ordtypelem2  7940  cantnfcl  8082  cantnflt  8087  cantnfp1lem3  8095  oemapvali  8099  cantnflem1c  8102  cantnflem1d  8103  cantnflem1  8104  cantnf  8108  cantnfclOLD  8112  cantnfltOLD  8117  cantnfp1lem3OLD  8121  cantnflem1cOLD  8125  cantnflem1dOLD  8126  cantnflem1OLD  8127  cantnfOLD  8130  cnfcom  8140  cnfcom3lem  8143  cnfcomOLD  8148  cnfcom3lemOLD  8151  infxpenlem  8387  ac10ct  8411  dfac12lem1  8519  dfac12lem2  8520  cfeq0  8632  cfsuc  8633  cff1  8634  cfflb  8635  cofsmo  8645  cfsmolem  8646  alephsing  8652  zorn2lem2  8873  ttukeylem3  8887  ttukeylem5  8889  ttukeylem6  8890  inar1  9149  predon  28850  soseq  28911  nobnddown  29038  nofulllem5  29043  ontgval  29473  aomclem6  30609
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