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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 3969 | . 2 ⊢ ¬ V ∈ V | |
2 | eleq1 2150 | . 2 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V)) | |
3 | 1, 2 | mtbiri 635 | 1 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1289 ∈ wcel 1438 Vcvv 2619 ∩ cint 3686 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-5 1381 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-ext 2070 ax-sep 3955 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-v 2621 |
This theorem is referenced by: (None) |
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