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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 3309 | . 2 ⊢ (𝐴 ∩ ∩ 𝑋) = (∩ 𝑋 ∩ 𝐴) | |
2 | intssuni2m 3842 | . . . 4 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ ∪ 𝒫 𝐴) | |
3 | ssid 3157 | . . . . 5 ⊢ 𝒫 𝐴 ⊆ 𝒫 𝐴 | |
4 | sspwuni 3944 | . . . . 5 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐴 ↔ ∪ 𝒫 𝐴 ⊆ 𝐴) | |
5 | 3, 4 | mpbi 144 | . . . 4 ⊢ ∪ 𝒫 𝐴 ⊆ 𝐴 |
6 | 2, 5 | sstrdi 3149 | . . 3 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → ∩ 𝑋 ⊆ 𝐴) |
7 | df-ss 3124 | . . 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 2209 | 1 ⊢ ((𝑋 ⊆ 𝒫 𝐴 ∧ ∃𝑥 𝑥 ∈ 𝑋) → (𝐴 ∩ ∩ 𝑋) = ∩ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∃wex 1479 ∈ wcel 2135 ∩ cin 3110 ⊆ wss 3111 𝒫 cpw 3553 ∪ cuni 3783 ∩ cint 3818 |
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-rex 2448 df-v 2723 df-in 3117 df-ss 3124 df-pw 3555 df-uni 3784 df-int 3819 |
This theorem is referenced by: (None) |
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