Theorem abnex 7007

Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 7008 and pwnex 7010. See the comment of abnexg 7006. (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 4829	. 2 ⊢ ¬ V ∈ V
2		alral 2957	. . 3 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ V (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹))
3		rexv 3251	. . . . . . 7 ⊢ (∃𝑥 ∈ V 𝑦 = 𝐹 ↔ ∃𝑥 𝑦 = 𝐹)
4	3	bicomi 214	. . . . . 6 ⊢ (∃𝑥 𝑦 = 𝐹 ↔ ∃𝑥 ∈ V 𝑦 = 𝐹)
5	4	abbii 2768	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹}
6	5	eleq1i 2721	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
7	6	biimpi 206	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
8		abnexg 7006	. . 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 190	1 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942  Vcvv 3231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-sn 4211  df-uni 4469  df-iun 4554

This theorem is referenced by:  snnex  7008  pwnex  7010