![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > intnexr | GIF version |
Description: If a class intersection is the universe, it is not a set. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Ref | Expression |
---|---|
intnexr | ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vprc 4147 | . 2 ⊢ ¬ V ∈ V | |
2 | eleq1 2250 | . 2 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V)) | |
3 | 1, 2 | mtbiri 676 | 1 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1363 ∈ wcel 2158 Vcvv 2749 ∩ cint 3856 |
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-in1 615 ax-in2 616 ax-5 1457 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-v 2751 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |