Theorem trssord 5958

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 5299	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴))
2		ordwe 5954	. . . . 5 ⊢ (Ord 𝐵 → E We 𝐵)
3	1, 2	impel 502	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴)
4	3	anim2i 611	. . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
5	4	3impb 1144	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6		df-ord 5944	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
7	5, 6	sylibr 226	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   ⊆ wss 3769  Tr wtr 4945   E cep 5224   We wwe 5270  Ord word 5940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-ral 3094  df-in 3776  df-ss 3783  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944

This theorem is referenced by:  ordin  5971  ssorduni  7219  suceloni  7247  ordom  7308  ordtypelem2  8666  hartogs  8691  card2on  8701  tskwe  9062  ondomon  9673  dford3lem2  38379  dford3  38380  iunord  43221