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| Mirrors > Home > MPE Home > Th. List > trel | Structured version Visualization version GIF version | ||
| Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) | 
| Ref | Expression | 
|---|---|
| trel | ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dftr2 5260 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴)) | |
| 2 | eleq12 2830 | . . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑦 ∈ 𝑥 ↔ 𝐵 ∈ 𝐶)) | |
| 3 | eleq1 2828 | . . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
| 4 | 3 | adantl 481 | . . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | 
| 5 | 2, 4 | anbi12d 632 | . . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))) | 
| 6 | eleq1 2828 | . . . . . 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 3599 | . . 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)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2107 Tr wtr 5258 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-uni 4907 df-tr 5259 | 
| This theorem is referenced by: trel3 5268 ordn2lp 6403 ordelord 6405 tz7.7 6409 ordtr1 6426 suctr 6469 trsuc 6470 trom 7897 elnn 7899 epfrs 9772 tcrank 9925 dfon2lem6 35790 tratrb 44561 truniALT 44566 onfrALTlem2 44571 trelded 44590 pwtrrVD 44850 suctrALT 44851 suctrALT2VD 44861 suctrALT2 44862 tratrbVD 44886 truniALTVD 44903 trintALTVD 44905 trintALT 44906 onfrALTlem2VD 44914 suctrALTcf 44947 suctrALTcfVD 44948 traxext 44999 modelac8prim 45014 | 
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