![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > intsng | 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 3546 | . . 3 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | inteqi 3783 | . 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴} |
3 | intprg 3812 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) | |
4 | 3 | anidms 395 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) |
5 | inidm 3290 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
6 | 4, 5 | eqtrdi 2189 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴) |
7 | 2, 6 | syl5eq 2185 | 1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ∩ cin 3075 {csn 3532 {cpr 3533 ∩ cint 3779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-v 2691 df-un 3080 df-in 3082 df-sn 3538 df-pr 3539 df-int 3780 |
This theorem is referenced by: intsn 3814 op1stbg 4408 riinint 4808 |
Copyright terms: Public domain | W3C validator |