Theorem trss 4137

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 2256	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2		sseq1 3203	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
3	1, 2	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴) ↔ (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)))
4	3	imbi2d 230	. . 3 ⊢ (𝑥 = 𝐵 → ((Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) ↔ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))))
5		dftr3 4132	. . . 4 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴)
6		rsp 2541	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
7	5, 6	sylbi 121	. . 3 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
8	4, 7	vtoclg 2821	. 2 ⊢ (𝐵 ∈ 𝐴 → (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)))
9	8	pm2.43b 52	1 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  ∀wral 2472   ⊆ wss 3154  Tr wtr 4128

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-in 3160  df-ss 3167  df-uni 3837  df-tr 4129

This theorem is referenced by:  trin  4138  triun  4141  trintssm  4144  tz7.2  4386  ordelss  4411  trsucss  4455  ordsucss  4537  ctinf  12590