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Theorem ordsson 7633
Description: Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
ordsson (Ord 𝐴𝐴 ⊆ On)

Proof of Theorem ordsson
StepHypRef Expression
1 ordon 7627 . 2 Ord On
2 ordeleqon 7632 . . . . 5 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
32biimpi 215 . . . 4 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
43adantr 481 . . 3 ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On))
5 ordsseleq 6295 . . 3 ((Ord 𝐴 ∧ Ord On) → (𝐴 ⊆ On ↔ (𝐴 ∈ On ∨ 𝐴 = On)))
64, 5mpbird 256 . 2 ((Ord 𝐴 ∧ Ord On) → 𝐴 ⊆ On)
71, 6mpan2 688 1 (Ord 𝐴𝐴 ⊆ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 844   = wceq 1539  wcel 2106  wss 3887  Ord word 6265  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:  onss  7634  orduni  7639  ordsucuniel  7671  ordsucuni  7676  iordsmo  8188  dfrecs3  8203  dfrecs3OLD  8204  tfr2b  8227  tz7.44-2  8238  ordiso2  9274  ordtypelem7  9283  ordtypelem8  9284  oiid  9300  r1tr  9534  r1ordg  9536  r1ord3g  9537  r1pwss  9542  r1val1  9544  rankwflemb  9551  r1elwf  9554  rankr1ai  9556  cflim2  10019  cfss  10021  cfslb  10022  cfslbn  10023  cfslb2n  10024  cofsmo  10025  coftr  10029  inaprc  10592  satfn  33317  dford5  33671  rdgprc  33770  nosepon  33868  limsucncmpi  34634
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