Theorem trssord 6397

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 5641	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴))
2		ordwe 6393	. . . . 5 ⊢ (Ord 𝐵 → E We 𝐵)
3	1, 2	impel 515	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴)
4	3	anim2i 629	. . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
5	4	3impb 1132	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6		df-ord 6383	. 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 401   ∧ w3a 1103   ⊆ wss 3900  Tr wtr 5215   E cep 5554   We wwe 5607  Ord word 6379

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ral 3081  df-ss 3901  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6383

This theorem is used by:  ordin  6411  ssorduni  7817  ordsuci  7846  ordom  7911  ordtypelem2  9540  hartogs  9565  card2on  9575  tskwe  10058  ondomon  10674  dford3lem2  44227  dford3  44228  iunord  51064