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Theorem abnex 7468
Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7469 and pwnex 7470. See the comment of abnexg 7467. (Contributed by BJ, 2-May-2021.)
Assertion
Ref Expression
abnex (∀𝑥(𝐹𝑉𝑥𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐹
Allowed substitution hints:   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem abnex
StepHypRef Expression
1 vprc 5210 . 2 ¬ V ∈ V
2 alral 3151 . . 3 (∀𝑥(𝐹𝑉𝑥𝐹) → ∀𝑥 ∈ V (𝐹𝑉𝑥𝐹))
3 rexv 3518 . . . . . . 7 (∃𝑥 ∈ V 𝑦 = 𝐹 ↔ ∃𝑥 𝑦 = 𝐹)
43bicomi 225 . . . . . 6 (∃𝑥 𝑦 = 𝐹 ↔ ∃𝑥 ∈ V 𝑦 = 𝐹)
54abbii 2883 . . . . 5 {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹}
65eleq1i 2900 . . . 4 ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
76biimpi 217 . . 3 ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
8 abnexg 7467 . . 3 (∀𝑥 ∈ V (𝐹𝑉𝑥𝐹) → ({𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V → V ∈ V))
92, 7, 8syl2im 40 . 2 (∀𝑥(𝐹𝑉𝑥𝐹) → ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → V ∈ V))
101, 9mtoi 200 1 (∀𝑥(𝐹𝑉𝑥𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1526   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  wrex 3136  Vcvv 3492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949  df-sn 4558  df-uni 4831  df-iun 4912
This theorem is referenced by:  snnex  7469  pwnex  7470
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