Theorem bj-xnex 32245

Description: Lemma for snnex 6862 and bj-pwnex 32246. (Contributed by BJ, 2-May-2021.)

Assertion

Ref	Expression
bj-xnex	⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∉ V)

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴

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

Proof of Theorem bj-xnex

Step	Hyp	Ref	Expression
1		nfa1 2015	. . . . 5 ⊢ Ⅎ𝑦∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴)
2		nfe1 2014	. . . . . . 7 ⊢ Ⅎ𝑦∃𝑦 𝑥 = 𝐴
3	2	nfab 2755	. . . . . 6 ⊢ Ⅎ𝑦{𝑥 ∣ ∃𝑦 𝑥 = 𝐴}
4	3	nfuni 4378	. . . . 5 ⊢ Ⅎ𝑦∪ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴}
5		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑦V
6		vex 3176	. . . . . . 7 ⊢ 𝑦 ∈ V
7	6	2a1i 12	. . . . . 6 ⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} → 𝑦 ∈ V))
8		nfcv 2751	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐴
9		nfv 1830	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
10		eleq2 2677	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐴))
11	10	biimprd 237	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝑥))
12		19.8a 2039	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
13	11, 12	jctird 565	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝐴 → (𝑦 ∈ 𝑥 ∧ ∃𝑦 𝑥 = 𝐴)))
14	8, 9, 13	spcimegf 3260	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → (𝑦 ∈ 𝐴 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∃𝑦 𝑥 = 𝐴)))
15	14	imp 444	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∃𝑦 𝑥 = 𝐴))
16	15	sps 2043	. . . . . . . 8 ⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∃𝑦 𝑥 = 𝐴))
17		eluniab 4383	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∃𝑦 𝑥 = 𝐴))
18	16, 17	sylibr 223	. . . . . . 7 ⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴})
19	18	a1d 25	. . . . . 6 ⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ V → 𝑦 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴}))
20	7, 19	impbid 201	. . . . 5 ⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → (𝑦 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ↔ 𝑦 ∈ V))
21	1, 4, 5, 20	eqrd 3586	. . . 4 ⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ∪ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} = V)
22		vprc 4724	. . . . 5 ⊢ ¬ V ∈ V
23	22	a1i 11	. . . 4 ⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ¬ V ∈ V)
24	21, 23	eqneltrd 2707	. . 3 ⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ¬ ∪ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∈ V)
25		uniexg 6853	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∈ V → ∪ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∈ V)
26	24, 25	nsyl 134	. 2 ⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∈ V)
27		df-nel 2783	. 2 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∈ V)
28	26, 27	sylibr 223	1 ⊢ (∀𝑦(𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∉ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ∉ wnel 2781  Vcvv 3173  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-uni 4373

This theorem is referenced by:  bj-pwnex  32246