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Theorem onss 6391
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 4716 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordsson 6390 . 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 1755    C_ wss 3316   Ord word 4705   Oncon0 4706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-tr 4374  df-eprel 4619  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710
This theorem is referenced by:  onuni  6393  onminex  6407  suceloni  6413  onssi  6437  tfi  6453  tfr3  6844  tz7.49  6886  tz7.49c  6887  oacomf1olem  6991  oeeulem  7028  ordtypelem2  7721  cantnfcl  7863  cantnflt  7868  cantnfp1lem3  7876  oemapvali  7880  cantnflem1c  7883  cantnflem1d  7884  cantnflem1  7885  cantnf  7889  cantnfclOLD  7893  cantnfltOLD  7898  cantnfp1lem3OLD  7902  cantnflem1cOLD  7906  cantnflem1dOLD  7907  cantnflem1OLD  7908  cantnfOLD  7911  cnfcom  7921  cnfcom3lem  7924  cnfcomOLD  7929  cnfcom3lemOLD  7932  infxpenlem  8168  ac10ct  8192  dfac12lem1  8300  dfac12lem2  8301  cfeq0  8413  cfsuc  8414  cff1  8415  cfflb  8416  cofsmo  8426  cfsmolem  8427  alephsing  8433  zorn2lem2  8654  ttukeylem3  8668  ttukeylem5  8670  ttukeylem6  8671  inar1  8930  predon  27501  soseq  27562  nobnddown  27689  nofulllem5  27694  ontgval  28125  aomclem6  29257
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