Theorem trssord 6401

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 6397	. . . . 5 ⊢ (Ord 𝐵 → E We 𝐵)
3	1, 2	impel 505	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴)
4	3	anim2i 617	. . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
5	4	3impb 1115	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6		df-ord 6387	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
7	5, 6	sylibr 234	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ⊆ wss 3951  Tr wtr 5259   E cep 5583   We wwe 5636  Ord word 6383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ral 3062  df-ss 3968  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387

This theorem is referenced by:  ordin  6414  ssorduni  7799  ordsuci  7828  sucexeloniOLD  7830  suceloniOLD  7832  ordom  7897  ordtypelem2  9559  hartogs  9584  card2on  9594  tskwe  9990  ondomon  10603  dford3lem2  43039  dford3  43040  iunord  49195