![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4138 | . . 3 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | inteqi 4414 | . 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴} |
3 | intprg 4446 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) | |
4 | 3 | anidms 675 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) |
5 | inidm 3784 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
6 | 4, 5 | syl6eq 2660 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴) |
7 | 2, 6 | syl5eq 2656 | 1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 {csn 4125 {cpr 4127 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-un 3545 df-in 3547 df-sn 4126 df-pr 4128 df-int 4411 |
This theorem is referenced by: intsn 4448 riinint 5303 elrfi 36275 |
Copyright terms: Public domain | W3C validator |