Theorem onelss 4482

Description: An element of an ordinal number is a subset of the number. (Contributed by NM, 5-Jun-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
onelss	⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Proof of Theorem onelss

Step	Hyp	Ref	Expression
1		eloni 4470	. 2 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		ordelss 4474	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
3	2	ex 115	. 2 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
4	1, 3	syl 14	1 ⊢ (𝐴 ∈ On → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2200   ⊆ wss 3198  Ord word 4457  Oncon0 4458

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-in 3204  df-ss 3211  df-uni 3892  df-tr 4186  df-iord 4461  df-on 4463

This theorem is referenced by:  onelssi  4524  ssorduni  4583  onsucelsucr  4604  tfisi  4683  tfrlem9  6480  nntri2or2  6661  phpelm  7048  exmidontri2or  7454  nninfctlemfo  12604  ennnfonelemk  13014