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Theorem abnex 7777
Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7778 and pwnex 7779. See the comment of abnexg 7776. (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 5315 . 2 ¬ V ∈ V
2 alral 3075 . . 3 (∀𝑥(𝐹𝑉𝑥𝐹) → ∀𝑥 ∈ V (𝐹𝑉𝑥𝐹))
3 rexv 3509 . . . . . . 7 (∃𝑥 ∈ V 𝑦 = 𝐹 ↔ ∃𝑥 𝑦 = 𝐹)
43bicomi 224 . . . . . 6 (∃𝑥 𝑦 = 𝐹 ↔ ∃𝑥 ∈ V 𝑦 = 𝐹)
54abbii 2809 . . . . 5 {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹}
65eleq1i 2832 . . . 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 7776 . . 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 1538   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-sn 4627  df-uni 4908  df-iun 4993
This theorem is referenced by:  snnex  7778  pwnex  7779
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