Theorem ordelss 4505

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 4504	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
2		trss 4222	. . 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 2205   ⊆ wss 3214  Tr wtr 4213  Ord word 4488

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-in 3220  df-ss 3227  df-uni 3920  df-tr 4214  df-iord 4492

This theorem is referenced by:  ordelord  4507  onelss  4513  ordsuc  4690  smores3  6537  tfrlem1  6552  tfrlemisucaccv  6569  tfrlemiubacc  6574  tfr1onlemsucaccv  6585  tfr1onlemubacc  6590  tfrcllemsucaccv  6598  tfrcllemubacc  6603  nntri1  6742  nnsseleq  6747  fict  7136  infnfi  7165  isinfinf  7167  ordiso2  7339  hashinfuni  11165