Theorem ordelord 4143

Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)

Assertion

Ref	Expression
ordelord	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Proof of Theorem ordelord

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2114	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2	1	anbi2d 445	. . . 4 ⊢ (𝑥 = 𝐵 → ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)))
3		ordeq 4134	. . . 4 ⊢ (𝑥 = 𝐵 → (Ord 𝑥 ↔ Ord 𝐵))
4	2, 3	imbi12d 227	. . 3 ⊢ (𝑥 = 𝐵 → (((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥) ↔ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)))
5		dford3 4129	. . . . . 6 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 Tr 𝑥))
6	5	simprbi 264	. . . . 5 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 Tr 𝑥)
7	6	r19.21bi 2422	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Tr 𝑥)
8		ordelss 4141	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝐴)
9		simpl 106	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝐴)
10		trssord 4142	. . . 4 ⊢ ((Tr 𝑥 ∧ 𝑥 ⊆ 𝐴 ∧ Ord 𝐴) → Ord 𝑥)
11	7, 8, 9, 10	syl3anc 1144	. . 3 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
12	4, 11	vtoclg 2628	. 2 ⊢ (𝐵 ∈ 𝐴 → ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵))
13	12	anabsi7 523	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Syntax hints:   → wi 4   ∧ wa 101   = wceq 1257   ∈ wcel 1407  ∀wral 2321   ⊆ wss 2942  Tr wtr 3879  Ord word 4124

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036

This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-in 2949  df-ss 2956  df-uni 3606  df-tr 3880  df-iord 4128

This theorem is referenced by:  tron  4144  ordelon  4145  ordsucg  4253  ordwe  4325  smores  5935