Theorem onss 4475

Description: An ordinal number is a subset of the class of ordinal numbers. (Contributed by NM, 5-Jun-1994.)

Assertion

Ref	Expression
onss	⊢ (𝐴 ∈ On → 𝐴 ⊆ On)

Proof of Theorem onss

Step	Hyp	Ref	Expression
1		eloni 4358	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordsson 4474	. 2 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
3	1, 2	syl 14	1 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)

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:  onuni  4476  onssi  4497  tfrexlem  6310  tfri3  6343  rdgivallem  6357  bj-omssonALT  13958