![]() |
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 4147 | . . 3 ⊢ ¬ V ∈ V | |
2 | inteq 3859 | . . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
3 | int0 3870 | . . . . 5 ⊢ ∩ ∅ = V | |
4 | 2, 3 | eqtrdi 2236 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
5 | 4 | eleq1d 2256 | . . 3 ⊢ (𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ V ∈ V)) |
6 | 1, 5 | mtbiri 676 | . 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝐴 ∈ V) |
7 | 6 | necon2ai 2411 | 1 ⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2158 ≠ wne 2357 Vcvv 2749 ∅c0 3434 ∩ 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-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 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-nfc 2318 df-ne 2358 df-ral 2470 df-v 2751 df-dif 3143 df-nul 3435 df-int 3857 |
This theorem is referenced by: fival 6983 |
Copyright terms: Public domain | W3C validator |