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| 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 4979 | . 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵}) | |
| 2 | intunsn.1 | . . . 4 ⊢ 𝐵 ∈ V | |
| 3 | 2 | intsn 4983 | . . 3 ⊢ ∩ {𝐵} = 𝐵 | 
| 4 | 3 | ineq2i 4216 | . 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵) | 
| 5 | 1, 4 | eqtri 2764 | 1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∪ cun 3948 ∩ cin 3949 {csn 4625 ∩ cint 4945 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-un 3955 df-in 3957 df-sn 4626 df-pr 4628 df-int 4946 | 
| This theorem is referenced by: fiint 9367 fiintOLD 9368 incexclem 15873 heibor1lem 37817 | 
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