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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 3869 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | trss 5155 | . . . . . 6 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
3 | trss 5155 | . . . . . 6 ⊢ (Tr 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵)) | |
4 | 2, 3 | im2anan9 623 | . . . . 5 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) |
5 | 1, 4 | syl5bi 245 | . . . 4 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) |
6 | ssin 4131 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
7 | 5, 6 | syl6ib 254 | . . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ⊆ (𝐴 ∩ 𝐵))) |
8 | 7 | ralrimiv 3094 | . 2 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥 ⊆ (𝐴 ∩ 𝐵)) |
9 | dftr3 5150 | . 2 ⊢ (Tr (𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
10 | 8, 9 | sylibr 237 | 1 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2112 ∀wral 3051 ∩ cin 3852 ⊆ wss 3853 Tr wtr 5146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-v 3400 df-in 3860 df-ss 3870 df-uni 4806 df-tr 5147 |
This theorem is referenced by: ordin 6221 tcmin 9335 ingru 10394 gruina 10397 dfon2lem4 33432 |
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