Theorem ordelss 4519

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 4518	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
2		trss 4233	. . 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 2209   ⊆ wss 3220  Tr wtr 4224  Ord word 4502

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-in 3226  df-ss 3233  df-uni 3931  df-tr 4225  df-iord 4506

This theorem is referenced by:  ordelord  4521  onelss  4527  ordsuc  4705  smores3  6554  tfrlem1  6569  tfrlemisucaccv  6586  tfrlemiubacc  6591  tfr1onlemsucaccv  6602  tfr1onlemubacc  6607  tfrcllemsucaccv  6615  tfrcllemubacc  6620  nntri1  6759  nnsseleq  6764  fict  7160  infnfi  7189  isinfinf  7191  ordiso2  7365  hashinfuni  11194