Theorem trssord 6395

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 5671	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴))
2		ordwe 6391	. . . . 5 ⊢ (Ord 𝐵 → E We 𝐵)
3	1, 2	impel 504	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴)
4	3	anim2i 615	. . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
5	4	3impb 1112	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6		df-ord 6381	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
7	5, 6	sylibr 233	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   ⊆ wss 3947  Tr wtr 5272   E cep 5587   We wwe 5638  Ord word 6377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-ral 3052  df-ss 3964  df-po 5596  df-so 5597  df-fr 5639  df-we 5641  df-ord 6381

This theorem is referenced by:  ordin  6408  ssorduni  7789  ordsuci  7819  sucexeloniOLD  7821  suceloniOLD  7823  ordom  7888  ordtypelem2  9564  hartogs  9589  card2on  9599  tskwe  9995  ondomon  10608  dford3lem2  42703  dford3  42704  iunord  48440