Theorem trss 3884

Description: An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)

Assertion

Ref	Expression
trss	⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Proof of Theorem trss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2141	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2		sseq1 3020	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
3	1, 2	imbi12d 232	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴) ↔ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)))
4	3	imbi2d 228	. . 3 ⊢ (𝑥 = 𝐵 → ((Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) ↔ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))))
5		dftr3 3879	. . . 4 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
6		rsp 2411	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
7	5, 6	sylbi 119	. . 3 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
8	4, 7	vtoclg 2658	. 2 ⊢ (𝐵 ∈ 𝐴 → (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)))
9	8	pm2.43b 51	1 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  ∀wral 2348   ⊆ wss 2973  Tr wtr 3875

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-tr 3876

This theorem is referenced by:  trin  3885  triun  3888  trintssm  3891  tz7.2  4109  ordelss  4134  trsucss  4178  ordsucss  4248