Theorem trssord 4365

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

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dford3 4352	. . . . . . 7 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 Tr 𝑥))
2	1	simprbi 273	. . . . . 6 ⊢ (Ord 𝐵 → ∀𝑥 ∈ 𝐵 Tr 𝑥)
3		ssralv 3211	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 Tr 𝑥 → ∀𝑥 ∈ 𝐴 Tr 𝑥))
4	2, 3	syl5 32	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (Ord 𝐵 → ∀𝑥 ∈ 𝐴 Tr 𝑥))
5	4	imp 123	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → ∀𝑥 ∈ 𝐴 Tr 𝑥)
6	5	anim2i 340	. . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
7	6	3impb 1194	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
8		dford3 4352	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
9	7, 8	sylibr 133	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973  ∀wral 2448   ⊆ wss 3121  Tr wtr 4087  Ord word 4347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-in 3127  df-ss 3134  df-iord 4351

This theorem is referenced by:  ordelord  4366  ordin  4370  ssorduni  4471  ordtriexmidlem  4503  ordtri2or2exmidlem  4510  onsucelsucexmidlem  4513  ordsuc  4547