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Theorem onss 4685
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 4505 . 2  |-  ( A  e.  On  ->  Ord  A )
2 ordsson 4684 . 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 1715    C_ wss 3238   Ord word 4494   Oncon0 4495
This theorem is referenced by:  onuni  4687  onminex  4701  suceloni  4707  onssi  4731  tfi  4747  tfr3  6557  tz7.49  6599  tz7.49c  6600  oacomf1olem  6704  oeeulem  6741  ordtypelem2  7381  cantnfcl  7515  cantnflt  7520  cantnfp1lem3  7529  oemapvali  7533  cantnflem1c  7536  cantnflem1d  7537  cantnflem1  7538  cantnf  7542  cnfcom  7550  cnfcom3lem  7553  infxpenlem  7788  ac10ct  7808  dfac12lem1  7916  dfac12lem2  7917  cfeq0  8029  cfsuc  8030  cff1  8031  cfflb  8032  cofsmo  8042  cfsmolem  8043  alephsing  8049  zorn2lem2  8271  ttukeylem3  8285  ttukeylem5  8287  ttukeylem6  8288  inar1  8544  predon  24934  soseq  24995  nobnddown  25096  nofulllem5  25101  ontgval  25612  aomclem6  26662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-tr 4216  df-eprel 4408  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499
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