Theorem ordelss 4526

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 4525	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
2		trss 4240	. . 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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∈ wcel 2209   ⊆ wss 3220  Tr wtr 4231  Ord word 4509

This proof depends on 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 proof 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 3936  df-tr 4232  df-iord 4513

This theorem is used by:  ordelord  4528  onelss  4534  ordsuc  4712  smores3  6568  tfrlem1  6583  tfrlemisucaccv  6600  tfrlemiubacc  6605  tfr1onlemsucaccv  6616  tfr1onlemubacc  6621  tfrcllemsucaccv  6629  tfrcllemubacc  6634  nntri1  6773  nnsseleq  6778  fict  7174  infnfi  7203  isinfinf  7205  ordiso2  7381  hashinfuni  11238