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Mirrors > Home > ILE Home > Th. List > rintm | GIF version |
Description: Relative intersection of an inhabited class. (Contributed by Jim Kingdon, 19-Aug-2018.) |
Ref | Expression |
---|---|
rintm | ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3176 | . 2 ⊢ (𝐴 ∩ ∩ 𝑋) = (∩ 𝑋 ∩ 𝐴) | |
2 | intssuni2m 3686 | . . . 4 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴) | |
3 | ssid 3029 | . . . . 5 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴 | |
4 | sspwuni 3786 | . . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴) | |
5 | 3, 4 | mpbi 143 | . . . 4 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴 |
6 | 2, 5 | syl6ss 3022 | . . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ 𝐴) |
7 | df-ss 2997 | . . 3 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (∩ 𝑋 ∩ 𝐴) = ∩ 𝑋) | |
8 | 6, 7 | sylib 120 | . 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (∩ 𝑋 ∩ 𝐴) = ∩ 𝑋) |
9 | 1, 8 | syl5eq 2127 | 1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∃wex 1422 ∈ wcel 1434 ∩ cin 2983 ⊆ wss 2984 𝒫 cpw 3406 ∪ cuni 3627 ∩ cint 3662 |
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 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-in 2990 df-ss 2997 df-pw 3408 df-uni 3628 df-int 3663 |
This theorem is referenced by: (None) |
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