Theorem ordelss 4474

Description: An element of an ordinal class is a subset of it. (Contributed by NM, 30-May-1994.)

Assertion

Ref	Expression
ordelss	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)

Proof of Theorem ordelss

Step	Hyp	Ref	Expression
1		ordtr 4473	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
2		trss 4194	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3	2	imp 124	. 2 ⊢ ((Tr 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
4	1, 3	sylan 283	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2200   ⊆ wss 3198  Tr wtr 4185  Ord word 4457

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-v 2802  df-in 3204  df-ss 3211  df-uni 3892  df-tr 4186  df-iord 4461

This theorem is referenced by:  ordelord  4476  onelss  4482  ordsuc  4659  smores3  6454  tfrlem1  6469  tfrlemisucaccv  6486  tfrlemiubacc  6491  tfr1onlemsucaccv  6502  tfr1onlemubacc  6507  tfrcllemsucaccv  6515  tfrcllemubacc  6520  nntri1  6659  nnsseleq  6664  fict  7050  infnfi  7077  isinfinf  7079  ordiso2  7225  hashinfuni  11029