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Mirrors > Home > MPE Home > Th. List > trin | Structured version Visualization version GIF version |
Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.) |
Ref | Expression |
---|---|
trin | ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3979 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | trss 5276 | . . . . . 6 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
3 | trss 5276 | . . . . . 6 ⊢ (Tr 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵)) | |
4 | 2, 3 | im2anan9 620 | . . . . 5 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) |
5 | 1, 4 | biimtrid 242 | . . . 4 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) |
6 | ssin 4247 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
7 | 5, 6 | imbitrdi 251 | . . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ⊆ (𝐴 ∩ 𝐵))) |
8 | 7 | ralrimiv 3143 | . 2 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥 ⊆ (𝐴 ∩ 𝐵)) |
9 | dftr3 5271 | . 2 ⊢ (Tr (𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
10 | 8, 9 | sylibr 234 | 1 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ∩ cin 3962 ⊆ wss 3963 Tr wtr 5265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 df-in 3970 df-ss 3980 df-uni 4913 df-tr 5266 |
This theorem is referenced by: ordin 6416 tcmin 9779 ingru 10853 gruina 10856 dfon2lem4 35768 |
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