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Mirrors > Home > MPE Home > Th. List > intsng | Structured version Visualization version GIF version |
Description: Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
intsng | ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4540 | . . 3 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | inteqi 4849 | . 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴} |
3 | intprg 4878 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) | |
4 | 3 | anidms 570 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) |
5 | inidm 4119 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
6 | 4, 5 | eqtrdi 2787 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴) |
7 | 2, 6 | syl5eq 2783 | 1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∩ cin 3852 {csn 4527 {cpr 4529 ∩ cint 4845 |
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 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-v 3400 df-un 3858 df-in 3860 df-sn 4528 df-pr 4530 df-int 4846 |
This theorem is referenced by: intsn 4883 riinint 5822 bj-snmoore 34968 bj-prmoore 34970 elrfi 40160 |
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