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Theorem ordsson 4245
 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 4147 . . 3 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
21ex 112 . 2 (Ord 𝐴 → (𝑥𝐴𝑥 ∈ On))
32ssrdv 2978 1 (Ord 𝐴𝐴 ⊆ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1409   ⊆ wss 2944  Ord word 4126  Oncon0 4127 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2951  df-ss 2958  df-uni 3608  df-tr 3882  df-iord 4130  df-on 4132 This theorem is referenced by:  onss  4246  orduni  4248  iordsmo  5942  tfrlemi14d  5977  ordiso2  6414
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