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Theorem abnex 7776
Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7777 and pwnex 7778. See the comment of abnexg 7775. (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 5321 . 2 ¬ V ∈ V
2 alral 3073 . . 3 (∀𝑥(𝐹𝑉𝑥𝐹) → ∀𝑥 ∈ V (𝐹𝑉𝑥𝐹))
3 rexv 3507 . . . . . . 7 (∃𝑥 ∈ V 𝑦 = 𝐹 ↔ ∃𝑥 𝑦 = 𝐹)
43bicomi 224 . . . . . 6 (∃𝑥 𝑦 = 𝐹 ↔ ∃𝑥 ∈ V 𝑦 = 𝐹)
54abbii 2807 . . . . 5 {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹}
65eleq1i 2830 . . . 4 ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
76biimpi 216 . . 3 ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
8 abnexg 7775 . . 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 199 1 (∀𝑥(𝐹𝑉𝑥𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-sn 4632  df-uni 4913  df-iun 4998
This theorem is referenced by:  snnex  7777  pwnex  7778
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