Theorem trel 5215

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 5209	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
2		eleq12 2827	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑦 ∈ 𝑥 ↔ 𝐵 ∈ 𝐶))
3		eleq1 2825	. . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
4	3	adantl 481	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
5	2, 4	anbi12d 633	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴)))
6		eleq1 2825	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
7	6	adantr 480	. . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
8	5, 7	imbi12d 344	. . . 4 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)))
9	8	spc2gv 3556	. . 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 217	1 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  Tr wtr 5207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-uni 4866  df-tr 5208

This theorem is referenced by:  trel3  5216  ordn2lp  6345  ordelord  6347  tz7.7  6351  ordtr1  6369  suctr  6413  trsuc  6414  trom  7827  elnn  7829  epfrs  9652  tcrank  9808  trssfir1om  35286  fineqvinfep  35300  trssfir1omregs  35311  dfon2lem6  35999  regsfromregtr  36687  tratrb  44886  truniALT  44891  onfrALTlem2  44896  trelded  44915  pwtrrVD  45174  suctrALT  45175  suctrALT2VD  45185  suctrALT2  45186  tratrbVD  45210  truniALTVD  45227  trintALTVD  45229  trintALT  45230  onfrALTlem2VD  45238  suctrALTcf  45271  suctrALTcfVD  45272  traxext  45327  modelac8prim  45342