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Theorem ordsson 7525
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 7519 . 2 Ord On
2 ordeleqon 7524 . . . . 5 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
32biimpi 219 . . . 4 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
43adantr 484 . . 3 ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On))
5 ordsseleq 6201 . . 3 ((Ord 𝐴 ∧ Ord On) → (𝐴 ⊆ On ↔ (𝐴 ∈ On ∨ 𝐴 = On)))
64, 5mpbird 260 . 2 ((Ord 𝐴 ∧ Ord On) → 𝐴 ⊆ On)
71, 6mpan2 691 1 (Ord 𝐴𝐴 ⊆ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 846   = wceq 1542  wcel 2114  wss 3843  Ord word 6171  Oncon0 6172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-11 2162  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-ord 6175  df-on 6176
This theorem is referenced by:  onss  7526  orduni  7530  ordsucuniel  7560  ordsucuni  7565  iordsmo  8025  dfrecs3  8040  tfr2b  8063  tz7.44-2  8074  ordiso2  9054  ordtypelem7  9063  ordtypelem8  9064  oiid  9080  r1tr  9280  r1ordg  9282  r1ord3g  9283  r1pwss  9288  r1val1  9290  rankwflemb  9297  r1elwf  9300  rankr1ai  9302  cflim2  9765  cfss  9767  cfslb  9768  cfslbn  9769  cfslb2n  9770  cofsmo  9771  coftr  9775  inaprc  10338  satfn  32890  dford5  33246  rdgprc  33346  nosepon  33513  limsucncmpi  34279
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