MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onssi Structured version   Visualization version   GIF version

Theorem onssi 7684
Description: An ordinal number is a subset of On. (Contributed by NM, 11-Aug-1994.)
Hypothesis
Ref Expression
onssi.1 𝐴 ∈ On
Assertion
Ref Expression
onssi 𝐴 ⊆ On

Proof of Theorem onssi
StepHypRef Expression
1 onssi.1 . 2 𝐴 ∈ On
2 onss 7634 . 2 (𝐴 ∈ On → 𝐴 ⊆ On)
31, 2ax-mp 5 1 𝐴 ⊆ On
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wss 3887  Oncon0 6266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270
This theorem is referenced by:  rankbnd2  9627  dfac12r  9902  cfsmolem  10026  ttukeylem6  10270
  Copyright terms: Public domain W3C validator