Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3300 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | trss 4083 | . . . . . 6 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴)) | |
3 | trss 4083 | . . . . . 6 ⊢ (Tr 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵)) | |
4 | 2, 3 | im2anan9 588 | . . . . 5 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) |
5 | 1, 4 | syl5bi 151 | . . . 4 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵))) |
6 | ssin 3339 | . . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
7 | 5, 6 | syl6ib 160 | . . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ⊆ (𝐴 ∩ 𝐵))) |
8 | 7 | ralrimiv 2536 | . 2 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥 ⊆ (𝐴 ∩ 𝐵)) |
9 | dftr3 4078 | . 2 ⊢ (Tr (𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥 ⊆ (𝐴 ∩ 𝐵)) | |
10 | 8, 9 | sylibr 133 | 1 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2135 ∀wral 2442 ∩ cin 3110 ⊆ wss 3111 Tr wtr 4074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-in 3117 df-ss 3124 df-uni 3784 df-tr 4075 |
This theorem is referenced by: ordin 4357 |
Copyright terms: Public domain | W3C validator |