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Mirrors > Home > ILE Home > Th. List > intss | GIF version |
Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.) |
Ref | Expression |
---|---|
intss | ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imim1 75 | . . . . 5 ⊢ ((𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ((𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))) | |
2 | 1 | al2imi 1392 | . . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦) → ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦))) |
3 | vex 2622 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | 3 | elint 3694 | . . . 4 ⊢ (𝑥 ∈ ∩ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑥 ∈ 𝑦)) |
5 | 3 | elint 3694 | . . . 4 ⊢ (𝑥 ∈ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)) |
6 | 2, 4, 5 | 3imtr4g 203 | . . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → (𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴)) |
7 | 6 | alrimiv 1802 | . 2 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵) → ∀𝑥(𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴)) |
8 | dfss2 3014 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵)) | |
9 | dfss2 3014 | . 2 ⊢ (∩ 𝐵 ⊆ ∩ 𝐴 ↔ ∀𝑥(𝑥 ∈ ∩ 𝐵 → 𝑥 ∈ ∩ 𝐴)) | |
10 | 7, 8, 9 | 3imtr4i 199 | 1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝐵 ⊆ ∩ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1287 ∈ wcel 1438 ⊆ wss 2999 ∩ cint 3688 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-in 3005 df-ss 3012 df-int 3689 |
This theorem is referenced by: (None) |
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