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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 4908 | . 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵}) | |
2 | intunsn.1 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 2 | intsn 4912 | . . 3 ⊢ ∩ {𝐵} = 𝐵 |
4 | 3 | ineq2i 4186 | . 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
5 | 1, 4 | eqtri 2844 | 1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ∩ cin 3935 {csn 4567 ∩ cint 4876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-un 3941 df-in 3943 df-sn 4568 df-pr 4570 df-int 4877 |
This theorem is referenced by: fiint 8795 incexclem 15191 heibor1lem 35102 |
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