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Theorem ordsson 4408
 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.)
Assertion
Ref Expression
ordsson (Ord 𝐴𝐴 ⊆ On)

Proof of Theorem ordsson
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ordelon 4305 . . 3 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
21ex 114 . 2 (Ord 𝐴 → (𝑥𝐴𝑥 ∈ On))
32ssrdv 3103 1 (Ord 𝐴𝐴 ⊆ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1480   ⊆ wss 3071  Ord word 4284  Oncon0 4285 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290 This theorem is referenced by:  onss  4409  orduni  4411  iordsmo  6194  tfrlemi14d  6230  tfr1onlemssrecs  6236  tfri1dALT  6248  tfrcllemssrecs  6249  ordiso2  6920
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