Theorem onelssi 4407

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 4365	. 2 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2136   ⊆ wss 3116  Oncon0 4341

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346

This theorem is referenced by:  onelini  4408  oneluni  4409  omp1eomlem  7059  enumctlemm  7079  ennnfonelemdc  12332  ennnfonelemg  12336  ctinfom  12361  2o01f  13876  isomninnlem  13909  iswomninnlem  13928  ismkvnnlem  13931