Theorem trssord 4335

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 4322	. . . . . . 7 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 Tr 𝑥))
2	1	simprbi 273	. . . . . 6 ⊢ (Ord 𝐵 → ∀𝑥 ∈ 𝐵 Tr 𝑥)
3		ssralv 3188	. . . . . 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 1178	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
8		dford3 4322	. 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 963  ∀wral 2432   ⊆ wss 3098  Tr wtr 4058  Ord word 4317

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-11 1483  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-ral 2437  df-in 3104  df-ss 3111  df-iord 4321

This theorem is referenced by:  ordelord  4336  ordin  4340  ssorduni  4440  ordtriexmidlem  4472  ordtri2or2exmidlem  4479  onsucelsucexmidlem  4482  ordsuc  4516