Theorem trssord 6377

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 5646	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴))
2		ordwe 6373	. . . . 5 ⊢ (Ord 𝐵 → E We 𝐵)
3	1, 2	impel 514	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴)
4	3	anim2i 628	. . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
5	4	3impb 1131	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6		df-ord 6363	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
7	5, 6	sylibr 237	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   ⊆ wss 3904  Tr wtr 5217   E cep 5559   We wwe 5612  Ord word 6359

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-ral 3079  df-ss 3921  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-ord 6363

This theorem is used by:  ordin  6391  ssorduni  7776  ordsuci  7805  ordom  7870  ordtypelem2  9479  hartogs  9504  card2on  9514  tskwe  9943  ondomon  10553  dford3lem2  43782  dford3  43783  iunord  50482