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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-vprc | GIF version |
Description: vprc 4135 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-vprc | ⊢ ¬ V ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nalset 14617 | . . 3 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
2 | vex 2740 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 2 | tbt 247 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
4 | 3 | albii 1470 | . . . . 5 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
5 | dfcleq 2171 | . . . . 5 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) | |
6 | 4, 5 | bitr4i 187 | . . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V) |
7 | 6 | exbii 1605 | . . 3 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V) |
8 | 1, 7 | mtbi 670 | . 2 ⊢ ¬ ∃𝑥 𝑥 = V |
9 | isset 2743 | . 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V) | |
10 | 8, 9 | mtbir 671 | 1 ⊢ ¬ V ∈ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 105 ∀wal 1351 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2737 |
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 614 ax-in2 615 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-13 2150 ax-14 2151 ax-ext 2159 ax-bdn 14539 ax-bdel 14543 ax-bdsep 14606 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-v 2739 |
This theorem is referenced by: bj-nvel 14619 bj-vnex 14620 bj-intexr 14630 bj-intnexr 14631 |
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