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Theorem ordsson 7497
 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 7491 . 2 Ord On
2 ordeleqon 7496 . . . . 5 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
32biimpi 217 . . . 4 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
43adantr 481 . . 3 ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On))
5 ordsseleq 6219 . . 3 ((Ord 𝐴 ∧ Ord On) → (𝐴 ⊆ On ↔ (𝐴 ∈ On ∨ 𝐴 = On)))
64, 5mpbird 258 . 2 ((Ord 𝐴 ∧ Ord On) → 𝐴 ⊆ On)
71, 6mpan2 687 1 (Ord 𝐴𝐴 ⊆ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 843   = wceq 1530   ∈ wcel 2107   ⊆ wss 3940  Ord word 6189  Oncon0 6190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7455 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-tr 5170  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6193  df-on 6194 This theorem is referenced by:  onss  7498  orduni  7502  ordsucuniel  7532  ordsucuni  7537  iordsmo  7990  dfrecs3  8005  tfr2b  8028  tz7.44-2  8039  ordiso2  8973  ordtypelem7  8982  ordtypelem8  8983  oiid  8999  r1tr  9199  r1ordg  9201  r1ord3g  9202  r1pwss  9207  r1val1  9209  rankwflemb  9216  r1elwf  9219  rankr1ai  9221  cflim2  9679  cfss  9681  cfslb  9682  cfslbn  9683  cfslb2n  9684  cofsmo  9685  coftr  9689  inaprc  10252  satfn  32505  dford5  32860  rdgprc  32942  nosepon  33075  limsucncmpi  33696
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