Theorem ordsson 4218

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 𝐴 → 𝐴 ⊆ On)

Proof of Theorem ordsson

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordelon 4120	. . 3 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
2	1	ex 108	. 2 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
3	2	ssrdv 2951	1 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)

Syntax hints:   → wi 4   ∈ wcel 1393   ⊆ wss 2917  Ord word 4099  Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105

This theorem is referenced by:  onss  4219  orduni  4221  iordsmo  5912  tfrlemi14d  5947  ordiso2  6357