Theorem trssord 6378

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 5645	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴))
2		ordwe 6374	. . . . 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 6364	. 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 3902  Tr wtr 5216   E cep 5558   We wwe 5611  Ord word 6360

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 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ral 3079  df-ss 3919  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-ord 6364

This theorem is used by:  ordin  6392  ssorduni  7781  ordsuci  7810  ordom  7875  ordtypelem2  9494  hartogs  9519  card2on  9529  tskwe  9958  ondomon  10574  dford3lem2  43870  dford3  43871  iunord  50604