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Theorem ordsson 7140
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 7133 . 2 Ord On
2 ordeleqon 7139 . . . . 5 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
32biimpi 206 . . . 4 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
43adantr 466 . . 3 ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On))
5 ordsseleq 5894 . . 3 ((Ord 𝐴 ∧ Ord On) → (𝐴 ⊆ On ↔ (𝐴 ∈ On ∨ 𝐴 = On)))
64, 5mpbird 247 . 2 ((Ord 𝐴 ∧ Ord On) → 𝐴 ⊆ On)
71, 6mpan2 671 1 (Ord 𝐴𝐴 ⊆ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 836   = wceq 1631  wcel 2145  wss 3723  Ord word 5864  Oncon0 5865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5868  df-on 5869
This theorem is referenced by:  onss  7141  orduni  7145  ordsucuniel  7175  ordsucuni  7180  iordsmo  7611  dfrecs3  7626  tfr2b  7649  tz7.44-2  7660  ordiso2  8580  ordtypelem7  8589  ordtypelem8  8590  oiid  8606  r1tr  8807  r1ordg  8809  r1ord3g  8810  r1pwss  8815  r1val1  8817  rankwflemb  8824  r1elwf  8827  rankr1ai  8829  cflim2  9291  cfss  9293  cfslb  9294  cfslbn  9295  cfslb2n  9296  cofsmo  9297  coftr  9301  inaprc  9864  dford5  31946  rdgprc  32036  nosepon  32155  limsucncmpi  32781
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