Theorem ordsson 4528

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 4418	. . 3 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
2	1	ex 115	. 2 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
3	2	ssrdv 3189	1 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)

Syntax hints:   → wi 4   ∈ wcel 2167   ⊆ wss 3157  Ord word 4397  Oncon0 4398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403

This theorem is referenced by:  onss  4529  orduni  4531  iordsmo  6355  tfrlemi14d  6391  tfr1onlemssrecs  6397  tfri1dALT  6409  tfrcllemssrecs  6410  ordiso2  7101