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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-vprc | GIF version |
Description: vprc 3909 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 10686 | . . 3 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
2 | vex 2604 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 2 | tbt 245 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
4 | 3 | albii 1399 | . . . . 5 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
5 | dfcleq 2075 | . . . . 5 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) | |
6 | 4, 5 | bitr4i 185 | . . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V) |
7 | 6 | exbii 1536 | . . 3 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V) |
8 | 1, 7 | mtbi 627 | . 2 ⊢ ¬ ∃𝑥 𝑥 = V |
9 | isset 2605 | . 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V) | |
10 | 8, 9 | mtbir 628 | 1 ⊢ ¬ V ∈ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 ∀wal 1282 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 ax-bdn 10608 ax-bdel 10612 ax-bdsep 10675 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-v 2603 |
This theorem is referenced by: bj-nvel 10688 bj-vnex 10689 bj-intexr 10699 bj-intnexr 10700 |
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