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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-intexr | GIF version |
Description: intexr 4136 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-intexr | ⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-vprc 13931 | . . 3 ⊢ ¬ V ∈ V | |
2 | inteq 3834 | . . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
3 | int0 3845 | . . . . 5 ⊢ ∩ ∅ = V | |
4 | 2, 3 | eqtrdi 2219 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
5 | 4 | eleq1d 2239 | . . 3 ⊢ (𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ V ∈ V)) |
6 | 1, 5 | mtbiri 670 | . 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝐴 ∈ V) |
7 | 6 | necon2ai 2394 | 1 ⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 Vcvv 2730 ∅c0 3414 ∩ cint 3831 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-bdn 13852 ax-bdel 13856 ax-bdsep 13919 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-v 2732 df-dif 3123 df-nul 3415 df-int 3832 |
This theorem is referenced by: (None) |
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