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Theorem onss 7268
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 5986 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ordsson 7267 . 2 (Ord 𝐴𝐴 ⊆ On)
31, 2syl 17 1 (𝐴 ∈ On → 𝐴 ⊆ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wss 3792  Ord word 5975  Oncon0 5976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-ord 5979  df-on 5980
This theorem is referenced by:  predon  7269  onuni  7271  onminex  7285  suceloni  7291  onssi  7315  tfi  7331  tfr3  7778  tz7.49  7823  tz7.49c  7824  oacomf1olem  7928  oeeulem  7965  ordtypelem2  8713  cantnfcl  8861  cantnflt  8866  cantnfp1lem3  8874  oemapvali  8878  cantnflem1c  8881  cantnflem1d  8882  cantnflem1  8883  cantnf  8887  cnfcom  8894  cnfcom3lem  8897  infxpenlem  9169  ac10ct  9190  dfac12lem1  9300  dfac12lem2  9301  cfeq0  9413  cfsuc  9414  cff1  9415  cfflb  9416  cofsmo  9426  cfsmolem  9427  alephsing  9433  zorn2lem2  9654  ttukeylem3  9668  ttukeylem5  9670  ttukeylem6  9671  inar1  9932  soseq  32343  nosupno  32438  ontgval  33013  aomclem6  38588
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