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Mirrors > Home > ILE Home > Th. List > trin | 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 3318 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | trss 4109 | . . . . . 6 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
3 | trss 4109 | . . . . . 6 ⊢ (Tr 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵)) | |
4 | 2, 3 | im2anan9 598 | . . . . 5 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) |
5 | 1, 4 | biimtrid 152 | . . . 4 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) |
6 | ssin 3357 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
7 | 5, 6 | syl6ib 161 | . . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ⊆ (𝐴 ∩ 𝐵))) |
8 | 7 | ralrimiv 2549 | . 2 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥 ⊆ (𝐴 ∩ 𝐵)) |
9 | dftr3 4104 | . 2 ⊢ (Tr (𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
10 | 8, 9 | sylibr 134 | 1 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 ∀wral 2455 ∩ cin 3128 ⊆ wss 3129 Tr wtr 4100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-in 3135 df-ss 3142 df-uni 3810 df-tr 4101 |
This theorem is referenced by: ordin 4384 |
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