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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 4000 | . . 3 ⊢ ¬ V ∈ V | |
2 | inteq 3721 | . . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
3 | int0 3732 | . . . . 5 ⊢ ∩ ∅ = V | |
4 | 2, 3 | syl6eq 2148 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
5 | 4 | eleq1d 2168 | . . 3 ⊢ (𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ V ∈ V)) |
6 | 1, 5 | mtbiri 641 | . 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝐴 ∈ V) |
7 | 6 | necon2ai 2321 | 1 ⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ∈ wcel 1448 ≠ wne 2267 Vcvv 2641 ∅c0 3310 ∩ cint 3718 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-v 2643 df-dif 3023 df-nul 3311 df-int 3719 |
This theorem is referenced by: (None) |
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