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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 4985 | . 2 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ ∩ {𝐵}) | |
2 | intunsn.1 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 2 | intsn 4989 | . . 3 ⊢ ∩ {𝐵} = 𝐵 |
4 | 3 | ineq2i 4225 | . 2 ⊢ (∩ 𝐴 ∩ ∩ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
5 | 1, 4 | eqtri 2763 | 1 ⊢ ∩ (𝐴 ∪ {𝐵}) = (∩ 𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ∩ cin 3962 {csn 4631 ∩ cint 4951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-un 3968 df-in 3970 df-sn 4632 df-pr 4634 df-int 4952 |
This theorem is referenced by: fiint 9364 fiintOLD 9365 incexclem 15869 heibor1lem 37796 |
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