Theorem abnex 4573

Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 4574 and pwnex 4575. See the comment of abnexg 4572. (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 4247	. 2 ⊢ ¬ V ∈ V
2		alral 2589	. . 3 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ∀𝑥 ∈ V (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹))
3		rexv 2834	. . . . . . 7 ⊢ (∃𝑥 ∈ V 𝑦 = 𝐹 ↔ ∃𝑥 𝑦 = 𝐹)
4	3	bicomi 132	. . . . . 6 ⊢ (∃𝑥 𝑦 = 𝐹 ↔ ∃𝑥 ∈ V 𝑦 = 𝐹)
5	4	abbii 2350	. . . . 5 ⊢ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹}
6	5	eleq1i 2300	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
7	6	biimpi 120	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
8		abnexg 4572	. . 3 ⊢ (∀𝑥 ∈ V (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V → V ∈ V))
9	2, 7, 8	syl2im 38	. 2 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → V ∈ V))
10	1, 9	mtoi 670	1 ⊢ (∀𝑥(𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  Vcvv 2815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-in 3220  df-ss 3227  df-sn 3700  df-uni 3920  df-iun 3998

This theorem is referenced by:  pwnex  4575