Theorem onelssi 4526

Description: A member of an ordinal number is a subset of it. (Contributed by NM, 11-Aug-1994.)

Hypothesis

Ref	Expression
on.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onelssi	⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)

Proof of Theorem onelssi

Step	Hyp	Ref	Expression
1		on.1	. 2 ⊢ 𝐴 ∈ On
2		onelss 4484	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2202   ⊆ wss 3200  Oncon0 4460

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-tr 4188  df-iord 4463  df-on 4465

This theorem is referenced by:  onelini  4527  oneluni  4528  omp1eomlem  7292  enumctlemm  7312  ennnfonelemdc  13019  ctinfom  13048  2o01f  16593  isomninnlem  16634  iswomninnlem  16653  ismkvnnlem  16656