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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 4449 | . . 3 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | inteqi 4750 | . 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴} |
3 | intprg 4780 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) | |
4 | 3 | anidms 559 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) |
5 | inidm 4077 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
6 | 4, 5 | syl6eq 2825 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴) |
7 | 2, 6 | syl5eq 2821 | 1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 ∩ cin 3823 {csn 4436 {cpr 4438 ∩ cint 4746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ral 3088 df-rab 3092 df-v 3412 df-un 3829 df-in 3831 df-sn 4437 df-pr 4439 df-int 4747 |
This theorem is referenced by: intsn 4782 riinint 5679 bj-snmoore 33949 elrfi 38720 |
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