Theorem trssord 5699

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 5061	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ( E We 𝐵 → E We 𝐴))
2		ordwe 5695	. . . . 5 ⊢ (Ord 𝐵 → E We 𝐵)
3	1, 2	impel 485	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → E We 𝐴)
4	3	anim2i 592	. . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ E We 𝐴))
5	4	3impb 1257	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ E We 𝐴))
6		df-ord 5685	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴))
7	5, 6	sylibr 224	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   ⊆ wss 3555  Tr wtr 4712   E cep 4983   We wwe 5032  Ord word 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-in 3562  df-ss 3569  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685

This theorem is referenced by:  ordin  5712  ssorduni  6932  suceloni  6960  ordom  7021  ordtypelem2  8368  hartogs  8393  card2on  8403  tskwe  8720  ondomon  9329  dford3lem2  37074  dford3  37075  iunord  41714