Theorem trssord 4401

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 4388	. . . . . . 7 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 Tr 𝑥))
2	1	simprbi 275	. . . . . 6 ⊢ (Ord 𝐵 → ∀𝑥 ∈ 𝐵 Tr 𝑥)
3		ssralv 3234	. . . . . 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 124	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → ∀𝑥 ∈ 𝐴 Tr 𝑥)
6	5	anim2i 342	. . 3 ⊢ ((Tr 𝐴 ∧ (𝐴 ⊆ 𝐵 ∧ Ord 𝐵)) → (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
7	6	3impb 1201	. 2 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
8		dford3 4388	. 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
9	7, 8	sylibr 134	1 ⊢ ((Tr 𝐴 ∧ 𝐴 ⊆ 𝐵 ∧ Ord 𝐵) → Ord 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980  ∀wral 2468   ⊆ wss 3144  Tr wtr 4119  Ord word 4383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-in 3150  df-ss 3157  df-iord 4387

This theorem is referenced by:  ordelord  4402  ordin  4406  ssorduni  4507  ordtriexmidlem  4539  ordtri2or2exmidlem  4546  onsucelsucexmidlem  4549  ordsuc  4583