Theorem trel 4952

Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
trel	⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))

Proof of Theorem trel

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr2 4947	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
2		eleq12 2868	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑦 ∈ 𝑥 ↔ 𝐵 ∈ 𝐶))
3		eleq1 2866	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
4	3	adantl 474	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
5	2, 4	anbi12d 625	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
6		eleq1 2866	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
7	6	adantr 473	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
8	5, 7	imbi12d 336	. . . 4 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)))
9	8	spc2gv 3484	. . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)))
10	9	pm2.43b 55	. 2 ⊢ (∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))
11	1, 10	sylbi 209	1 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385  ∀wal 1651   = wceq 1653   ∈ wcel 2157  Tr wtr 4945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ss 3783  df-uni 4629  df-tr 4946

This theorem is referenced by:  trel3  4953  ordn2lp  5961  ordelord  5963  tz7.7  5967  ordtr1  5984  suctr  6024  trsuc  6025  ordom  7308  elnn  7309  epfrs  8857  tcrank  8997  dfon2lem6  32205  tratrb  39522  truniALT  39527  onfrALTlem2  39532  trelded  39551  pwtrrVD  39821  suctrALT  39822  suctrALT2VD  39832  suctrALT2  39833  tratrbVD  39857  truniALTVD  39874  trintALTVD  39876  trintALT  39877  onfrALTlem2VD  39885  suctrALTcf  39918  suctrALTcfVD  39919