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Theorem bj-vprc 12928
Description: vprc 4028 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-vprc ¬ V ∈ V

Proof of Theorem bj-vprc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-nalset 12927 . . 3 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 2661 . . . . . . 7 𝑦 ∈ V
32tbt 246 . . . . . 6 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1429 . . . . 5 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2109 . . . . 5 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 186 . . . 4 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1567 . . 3 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 642 . 2 ¬ ∃𝑥 𝑥 = V
9 isset 2664 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
108, 9mtbir 643 1 ¬ V ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104  wal 1312   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658
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 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097  ax-bdn 12849  ax-bdel 12853  ax-bdsep 12916
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660
This theorem is referenced by:  bj-nvel  12929  bj-vnex  12930  bj-intexr  12940  bj-intnexr  12941
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