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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-intexr | GIF version |
Description: intexr 4112 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 13513 | . . 3 ⊢ ¬ V ∈ V | |
2 | inteq 3811 | . . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
3 | int0 3822 | . . . . 5 ⊢ ∩ ∅ = V | |
4 | 2, 3 | eqtrdi 2206 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
5 | 4 | eleq1d 2226 | . . 3 ⊢ (𝐴 = ∅ → (∩ 𝐴 ∈ V ↔ V ∈ V)) |
6 | 1, 5 | mtbiri 665 | . 2 ⊢ (𝐴 = ∅ → ¬ ∩ 𝐴 ∈ V) |
7 | 6 | necon2ai 2381 | 1 ⊢ (∩ 𝐴 ∈ V → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 ≠ wne 2327 Vcvv 2712 ∅c0 3394 ∩ cint 3808 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-bdn 13434 ax-bdel 13438 ax-bdsep 13501 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-v 2714 df-dif 3104 df-nul 3395 df-int 3809 |
This theorem is referenced by: (None) |
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