Theorem abnex 7792

Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7793 and pwnex 7794. See the comment of abnexg 7791. (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 5333	. 2 ⊢ ¬ V ∈ V
2		alral 3081	. . 3 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ V (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹))
3		rexv 3517	. . . . . . 7 ⊢ (∃𝑥 ∈ V 𝑦 = 𝐹 ↔ ∃𝑥 𝑦 = 𝐹)
4	3	bicomi 224	. . . . . 6 ⊢ (∃𝑥 𝑦 = 𝐹 ↔ ∃𝑥 ∈ V 𝑦 = 𝐹)
5	4	abbii 2812	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹}
6	5	eleq1i 2835	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
7	6	biimpi 216	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
8		abnexg 7791	. . 3 ⊢ (∀𝑥 ∈ V (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V → V ∈ V))
9	2, 7, 8	syl2im 40	. 2 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → V ∈ V))
10	1, 9	mtoi 199	1 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-sn 4649  df-uni 4932  df-iun 5017

This theorem is referenced by:  snnex  7793  pwnex  7794