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Theorem onss 6625
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 4897 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordsson 6624 . 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 1819    C_ wss 3471   Ord word 4886   Oncon0 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891
This theorem is referenced by:  onuni  6627  onminex  6641  suceloni  6647  onssi  6671  tfi  6687  tfr3  7086  tz7.49  7128  tz7.49c  7129  oacomf1olem  7231  oeeulem  7268  ordtypelem2  7962  cantnfcl  8103  cantnflt  8108  cantnfp1lem3  8116  oemapvali  8120  cantnflem1c  8123  cantnflem1d  8124  cantnflem1  8125  cantnf  8129  cantnfclOLD  8133  cantnfltOLD  8138  cantnfp1lem3OLD  8142  cantnflem1cOLD  8146  cantnflem1dOLD  8147  cantnflem1OLD  8148  cantnfOLD  8151  cnfcom  8161  cnfcom3lem  8164  cnfcomOLD  8169  cnfcom3lemOLD  8172  infxpenlem  8408  ac10ct  8432  dfac12lem1  8540  dfac12lem2  8541  cfeq0  8653  cfsuc  8654  cff1  8655  cfflb  8656  cofsmo  8666  cfsmolem  8667  alephsing  8673  zorn2lem2  8894  ttukeylem3  8908  ttukeylem5  8910  ttukeylem6  8911  inar1  9170  predon  29469  soseq  29530  nobnddown  29657  nofulllem5  29662  ontgval  30080  aomclem6  31188
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