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 5647	. . . . 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 1130	. 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)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   ⊆ wss 3904  Tr wtr 5217   E cep 5560   We wwe 5613  Ord word 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-ral 3078  df-ss 3921  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363

This theorem is referenced by:  ordin  6391  ssorduni  7777  ordsuci  7806  ordom  7871  ordtypelem2  9480  hartogs  9505  card2on  9515  tskwe  9935  ondomon  10546  dford3lem2  43724  dford3  43725  iunord  50421