![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3232 | . 2 ⊢ (𝐴 ∩ ∩ 𝑋) = (∩ 𝑋 ∩ 𝐴) | |
2 | intssuni2m 3759 | . . . 4 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴) | |
3 | ssid 3081 | . . . . 5 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴 | |
4 | sspwuni 3861 | . . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴) | |
5 | 3, 4 | mpbi 144 | . . . 4 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴 |
6 | 2, 5 | syl6ss 3073 | . . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ 𝐴) |
7 | df-ss 3048 | . . 3 ⊢ (∩ 𝑋 ⊆ 𝐴 ↔ (∩ 𝑋 ∩ 𝐴) = ∩ 𝑋) | |
8 | 6, 7 | sylib 121 | . 2 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (∩ 𝑋 ∩ 𝐴) = ∩ 𝑋) |
9 | 1, 8 | syl5eq 2157 | 1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1312 ∃wex 1449 ∈ wcel 1461 ∩ cin 3034 ⊆ wss 3035 𝒫 cpw 3474 ∪ cuni 3700 ∩ cint 3735 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-in 3041 df-ss 3048 df-pw 3476 df-uni 3701 df-int 3736 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |