Theorem trss 3863

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 2100	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2		sseq1 2966	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
3	1, 2	imbi12d 223	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴) ↔ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)))
4	3	imbi2d 219	. . 3 ⊢ (𝑥 = 𝐵 → ((Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) ↔ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))))
5		dftr3 3858	. . . 4 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
6		rsp 2369	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
7	5, 6	sylbi 114	. . 3 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
8	4, 7	vtoclg 2613	. 2 ⊢ (𝐵 ∈ 𝐴 → (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)))
9	8	pm2.43b 46	1 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ⊆ wss 2917  Tr wtr 3854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855

This theorem is referenced by:  trin  3864  triun  3867  trint0m  3871  tz7.2  4091  ordelss  4116  trsucss  4160  ordsucss  4230