![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > intsn | GIF version |
Description: The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.) |
Ref | Expression |
---|---|
intsn.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
intsn | ⊢ ∩ {𝐴} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | intsng 3900 | . 2 ⊢ (𝐴 ∈ V → ∩ {𝐴} = 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∩ {𝐴} = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 Vcvv 2756 {csn 3614 ∩ cint 3866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2758 df-un 3153 df-in 3155 df-sn 3620 df-pr 3621 df-int 3867 |
This theorem is referenced by: uniintsnr 3902 intunsn 3904 op1stb 4503 op2ndb 5137 ssfii 7019 |
Copyright terms: Public domain | W3C validator |