Theorem abnex 7755

Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7756 and pwnex 7757. See the comment of abnexg 7754. (Contributed by BJ, 2-May-2021.)

Assertion

Ref	Expression
abnex	⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐹

Allowed substitution hints:   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem abnex

Step	Hyp	Ref	Expression
1		vprc 5285	. 2 ⊢ ¬ V ∈ V
2		alral 3100	. . 3 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ V (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹))
3		rexv 3490	. . . . . . 7 ⊢ (∃𝑥 ∈ V 𝑦 = 𝐹 ↔ ∃𝑥 𝑦 = 𝐹)
4	3	bicomi 227	. . . . . 6 ⊢ (∃𝑥 𝑦 = 𝐹 ↔ ∃𝑥 ∈ V 𝑦 = 𝐹)
5	4	abbii 2836	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹}
6	5	eleq1i 2860	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
7	6	biimpi 219	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
8		abnexg 7754	. . 3 ⊢ (∀𝑥 ∈ V (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V → V ∈ V))
9	2, 7, 8	syl2im 41	. 2 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → V ∈ V))
10	1, 9	mtoi 202	1 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095  Vcvv 3463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-in 3920  df-ss 3930  df-sn 4595  df-uni 4877  df-iun 4962

This theorem is referenced by:  snnex  7756  pwnex  7757