Theorem ordsson 4619

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 4509	. . 3 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
2	1	ex 115	. 2 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ On))
3	2	ssrdv 3248	1 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)

Syntax hints:   → wi 4   ∈ wcel 2205   ⊆ wss 3214  Ord word 4488  Oncon0 4489

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-in 3220  df-ss 3227  df-uni 3920  df-tr 4214  df-iord 4492  df-on 4494

This theorem is referenced by:  onss  4620  orduni  4622  iordsmo  6541  tfrlemi14d  6577  tfr1onlemssrecs  6583  tfri1dALT  6595  tfrcllemssrecs  6596  ordiso2  7339