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Theorem ordsson 4474
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 4366 . . 3 ((Ord 𝐴𝑥𝐴) → 𝑥 ∈ On)
21ex 114 . 2 (Ord 𝐴 → (𝑥𝐴𝑥 ∈ On))
32ssrdv 3153 1 (Ord 𝐴𝐴 ⊆ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2141  wss 3121  Ord word 4345  Oncon0 4346
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-tr 4086  df-iord 4349  df-on 4351
This theorem is referenced by:  onss  4475  orduni  4477  iordsmo  6273  tfrlemi14d  6309  tfr1onlemssrecs  6315  tfri1dALT  6327  tfrcllemssrecs  6328  ordiso2  7008
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