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Mirrors > Home > MPE Home > Th. List > trel3 | Structured version Visualization version GIF version |
Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) |
Ref | Expression |
---|---|
trel3 | ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐵 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anass 1096 | . . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) ↔ (𝐵 ∈ 𝐶 ∧ (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴))) | |
2 | trel 5235 | . . . 4 ⊢ (Tr 𝐴 → ((𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐶 ∈ 𝐴)) | |
3 | 2 | anim2d 613 | . . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ (𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴)) → (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))) |
4 | 1, 3 | biimtrid 241 | . 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → (𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴))) |
5 | trel 5235 | . 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | |
6 | 4, 5 | syld 47 | 1 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐶 ∧ 𝐶 ∈ 𝐷 ∧ 𝐷 ∈ 𝐴) → 𝐵 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 Tr wtr 5226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 df-in 3921 df-ss 3931 df-uni 4870 df-tr 5227 |
This theorem is referenced by: ordelord 6343 |
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