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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 4639 | . . 3 ⊢ {𝐴} = {𝐴, 𝐴} | |
| 2 | 1 | inteqi 4950 | . 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴} |
| 3 | intprg 4981 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) | |
| 4 | 3 | anidms 566 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) |
| 5 | inidm 4227 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
| 6 | 4, 5 | eqtrdi 2793 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴) |
| 7 | 2, 6 | eqtrid 2789 | 1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 {csn 4626 {cpr 4628 ∩ cint 4946 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-in 3958 df-sn 4627 df-pr 4629 df-int 4947 |
| This theorem is referenced by: intsn 4984 riinint 5982 bj-snmoore 37114 bj-prmoore 37116 elrfi 42705 |
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