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Theorem onss 7032
Description: An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
onss (𝐴 ∈ On → 𝐴 ⊆ On)

Proof of Theorem onss
StepHypRef Expression
1 eloni 5771 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordsson 7031 . 2 (Ord 𝐴𝐴 ⊆ On)
31, 2syl 17 1 (𝐴 ∈ On → 𝐴 ⊆ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  wss 3607  Ord word 5760  Oncon0 5761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765
This theorem is referenced by:  predon  7033  onuni  7035  onminex  7049  suceloni  7055  onssi  7079  tfi  7095  tfr3  7540  tz7.49  7585  tz7.49c  7586  oacomf1olem  7689  oeeulem  7726  ordtypelem2  8465  cantnfcl  8602  cantnflt  8607  cantnfp1lem3  8615  oemapvali  8619  cantnflem1c  8622  cantnflem1d  8623  cantnflem1  8624  cantnf  8628  cnfcom  8635  cnfcom3lem  8638  infxpenlem  8874  ac10ct  8895  dfac12lem1  9003  dfac12lem2  9004  cfeq0  9116  cfsuc  9117  cff1  9118  cfflb  9119  cofsmo  9129  cfsmolem  9130  alephsing  9136  zorn2lem2  9357  ttukeylem3  9371  ttukeylem5  9373  ttukeylem6  9374  inar1  9635  soseq  31879  nosupno  31974  ontgval  32555  aomclem6  37946
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