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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 5176 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴)) | |
2 | eleq12 2904 | . . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑦 ∈ 𝑥 ↔ 𝐵 ∈ 𝐶)) | |
3 | eleq1 2902 | . . . . . . 7 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴)) | |
4 | 3 | adantl 484 | . . . . . 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 2902 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) | |
7 | 6 | adantr 483 | . . . . 5 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (𝑦 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)) |
8 | 5, 7 | imbi12d 347 | . . . 4 ⊢ ((𝑦 = 𝐵 ∧ 𝑥 = 𝐶) → (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴))) |
9 | 8 | spc2gv 3603 | . . 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 219 | 1 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Tr wtr 5174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-in 3945 df-ss 3954 df-uni 4841 df-tr 5175 |
This theorem is referenced by: trel3 5182 ordn2lp 6213 ordelord 6215 tz7.7 6219 ordtr1 6236 suctr 6276 trsuc 6277 ordom 7591 elnn 7592 epfrs 9175 tcrank 9315 dfon2lem6 33035 tratrb 40877 truniALT 40882 onfrALTlem2 40887 trelded 40906 pwtrrVD 41166 suctrALT 41167 suctrALT2VD 41177 suctrALT2 41178 tratrbVD 41202 truniALTVD 41219 trintALTVD 41221 trintALT 41222 onfrALTlem2VD 41230 suctrALTcf 41263 suctrALTcfVD 41264 |
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