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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-intexr | GIF version |
Description: intexr 3895 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 9351 | . . 3 ⊢ ¬ V ∈ V | |
2 | inteq 3609 | . . . . 5 ⊢ (A = ∅ → ∩ A = ∩ ∅) | |
3 | int0 3620 | . . . . 5 ⊢ ∩ ∅ = V | |
4 | 2, 3 | syl6eq 2085 | . . . 4 ⊢ (A = ∅ → ∩ A = V) |
5 | 4 | eleq1d 2103 | . . 3 ⊢ (A = ∅ → (∩ A ∈ V ↔ V ∈ V)) |
6 | 1, 5 | mtbiri 599 | . 2 ⊢ (A = ∅ → ¬ ∩ A ∈ V) |
7 | 6 | necon2ai 2253 | 1 ⊢ (∩ A ∈ V → A ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 ≠ wne 2201 Vcvv 2551 ∅c0 3218 ∩ cint 3606 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-bdn 9272 ax-bdel 9276 ax-bdsep 9339 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-v 2553 df-dif 2914 df-nul 3219 df-int 3607 |
This theorem is referenced by: (None) |
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