Theorem trssord 6352

Description: A transitive subclass of an ordinal class is ordinal. (Contributed by NM, 29-May-1994.)

Assertion

Ref	Expression
trssord	⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Proof of Theorem trssord

Step	Hyp	Ref	Expression
1		wess 5626	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴))
2		ordwe 6348	. . . . 5 ⊢ (Ord 𝐵 → E We 𝐵)
3	1, 2	impel 512	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴)
4	3	anim2i 625	. . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
5	4	3impb 1123	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6		df-ord 6338	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
7	5, 6	sylibr 236	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   ⊆ wss 3899  Tr wtr 5201   E cep 5539   We wwe 5592  Ord word 6334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1097  df-ral 3071  df-ss 3916  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-ord 6338

This theorem is referenced by:  ordin  6365  ssorduni  7751  ordsuci  7780  ordom  7845  ordtypelem2  9457  hartogs  9482  card2on  9492  tskwe  9898  ondomon  10510  dford3lem2  43552  dford3  43553  iunord  50245