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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-vprc | GIF version |
Description: vprc 4068 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 13264 | . . 3 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
2 | vex 2692 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 2 | tbt 246 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
4 | 3 | albii 1447 | . . . . 5 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
5 | dfcleq 2134 | . . . . 5 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) | |
6 | 4, 5 | bitr4i 186 | . . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V) |
7 | 6 | exbii 1585 | . . 3 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V) |
8 | 1, 7 | mtbi 660 | . 2 ⊢ ¬ ∃𝑥 𝑥 = V |
9 | isset 2695 | . 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V) | |
10 | 8, 9 | mtbir 661 | 1 ⊢ ¬ V ∈ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 ∀wal 1330 = wceq 1332 ∃wex 1469 ∈ wcel 1481 Vcvv 2689 |
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-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-ext 2122 ax-bdn 13186 ax-bdel 13190 ax-bdsep 13253 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 |
This theorem is referenced by: bj-nvel 13266 bj-vnex 13267 bj-intexr 13277 bj-intnexr 13278 |
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