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Theorem onss 4582
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 4402 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordsson 4581 . 2  |-  ( Ord 
A  ->  A  C_  On )
31, 2syl 15 1  |-  ( A  e.  On  ->  A  C_  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    C_ wss 3152   Ord word 4391   Oncon0 4392
This theorem is referenced by:  onuni  4584  onminex  4598  suceloni  4604  onssi  4628  tfi  4644  tfr3  6415  tz7.49  6457  tz7.49c  6458  oacomf1olem  6562  oeeulem  6599  ordtypelem2  7234  cantnfcl  7368  cantnflt  7373  cantnfp1lem3  7382  oemapvali  7386  cantnflem1c  7389  cantnflem1d  7390  cantnflem1  7391  cantnf  7395  cnfcom  7403  cnfcom3lem  7406  infxpenlem  7641  ac10ct  7661  dfac12lem1  7769  dfac12lem2  7770  cfeq0  7882  cfsuc  7883  cff1  7884  cfflb  7885  cofsmo  7895  cfsmolem  7896  alephsing  7902  zorn2lem2  8124  ttukeylem3  8138  ttukeylem5  8140  ttukeylem6  8141  inar1  8397  predon  24193  soseq  24254  nobnddown  24355  nofulllem5  24360  ontgval  24870  valdom  25884  eltintpar  25899  inttaror  25900  aomclem6  27156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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