Theorem ordelss 4434

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 4433	. 2 ⊢ (Ord 𝐴 → Tr 𝐴)
2		trss 4159	. . 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 2177   ⊆ wss 3170  Tr wtr 4150  Ord word 4417

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-in 3176  df-ss 3183  df-uni 3857  df-tr 4151  df-iord 4421

This theorem is referenced by:  ordelord  4436  onelss  4442  ordsuc  4619  smores3  6392  tfrlem1  6407  tfrlemisucaccv  6424  tfrlemiubacc  6429  tfr1onlemsucaccv  6440  tfr1onlemubacc  6445  tfrcllemsucaccv  6453  tfrcllemubacc  6458  nntri1  6595  nnsseleq  6600  fict  6980  infnfi  7007  isinfinf  7009  ordiso2  7152  hashinfuni  10944