Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > intunsn | Structured version Visualization version 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 4906 | . 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵}) | |
2 | intunsn.1 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 2 | intsn 4912 | . . 3 ⊢ ∩ {𝐵} = 𝐵 |
4 | 3 | ineq2i 4139 | . 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
5 | 1, 4 | eqtri 2766 | 1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2111 Vcvv 3421 ∪ cun 3879 ∩ cin 3880 {csn 4556 ∩ cint 4874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rab 3071 df-v 3423 df-un 3886 df-in 3888 df-sn 4557 df-pr 4559 df-int 4875 |
This theorem is referenced by: fiint 8973 incexclem 15428 heibor1lem 35734 |
Copyright terms: Public domain | W3C validator |