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Mirrors > Home > ILE Home > Th. List > intunsn | GIF version |
Description: Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.) |
Ref | Expression |
---|---|
intunsn.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
intunsn | ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intun 3797 | . 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵}) | |
2 | intunsn.1 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 2 | intsn 3801 | . . 3 ⊢ ∩ {𝐵} = 𝐵 |
4 | 3 | ineq2i 3269 | . 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
5 | 1, 4 | eqtri 2158 | 1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2681 ∪ cun 3064 ∩ cin 3065 {csn 3522 ∩ cint 3766 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-un 3070 df-in 3072 df-sn 3528 df-pr 3529 df-int 3767 |
This theorem is referenced by: fiintim 6810 |
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