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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 4984 | . 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵}) | |
2 | intunsn.1 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 2 | intsn 4990 | . . 3 ⊢ ∩ {𝐵} = 𝐵 |
4 | 3 | ineq2i 4209 | . 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
5 | 1, 4 | eqtri 2759 | 1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∪ cun 3946 ∩ cin 3947 {csn 4628 ∩ cint 4950 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-un 3953 df-in 3955 df-sn 4629 df-pr 4631 df-int 4951 |
This theorem is referenced by: fiint 9330 incexclem 15789 heibor1lem 37143 |
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