Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > intexr | GIF version |
Description: If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Ref | Expression |
---|---|
intexr | ⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vprc 4060 | . . 3 ⊢ ¬ V ∈ V | |
2 | inteq 3774 | . . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
3 | int0 3785 | . . . . 5 ⊢ ∩ ∅ = V | |
4 | 2, 3 | syl6eq 2188 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
5 | 4 | eleq1d 2208 | . . 3 ⊢ (𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ V ∈ V)) |
6 | 1, 5 | mtbiri 664 | . 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝐴 ∈ V) |
7 | 6 | necon2ai 2362 | 1 ⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 Vcvv 2686 ∅c0 3363 ∩ cint 3771 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-v 2688 df-dif 3073 df-nul 3364 df-int 3772 |
This theorem is referenced by: fival 6858 |
Copyright terms: Public domain | W3C validator |