Theorem eliin2f 44501

Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
eliin2f.1	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
eliin2f	⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem eliin2f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliin 5005	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
2	1	adantl 480	. 2 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ∈ V) → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3		prcnel 3497	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
4	3	adantl 480	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
5		n0 4350	. . . . . . . . 9 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
6	5	biimpi 215	. . . . . . . 8 ⊢ (𝐵 ≠ ∅ → ∃𝑦 𝑦 ∈ 𝐵)
7	6	adantr 479	. . . . . . 7 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 𝑦 ∈ 𝐵)
8		prcnel 3497	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
9	8	a1d 25	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ V → (𝑦 ∈ 𝐵 → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
10	9	adantl 480	. . . . . . . . 9 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦 ∈ 𝐵 → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
11	10	ancld 549	. . . . . . . 8 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)))
12	11	eximdv 1912	. . . . . . 7 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (∃𝑦 𝑦 ∈ 𝐵 → ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)))
13	7, 12	mpd 15	. . . . . 6 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
14		df-rex 3068	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
15	13, 14	sylibr 233	. . . . 5 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
16		eliin2f.1	. . . . . 6 ⊢ Ⅎ𝑥𝐵
17		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑦𝐵
18		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑦 ¬ 𝐴 ∈ 𝐶
19		nfcsb1v 3919	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
20	19	nfel2 2918	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶
21	20	nfn 1852	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶
22		csbeq1a 3908	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
23	22	eleq2d 2815	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
24	23	notbid 317	. . . . . 6 ⊢ (𝑥 = 𝑦 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
25	16, 17, 18, 21, 24	cbvrexfw 3300	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
26	15, 25	sylibr 233	. . . 4 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶)
27		rexnal 3097	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶 ↔ ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
28	26, 27	sylib 217	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
29	4, 28	2falsed 375	. 2 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
30	2, 29	pm2.61dan 811	1 ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394  ∃wex 1773   ∈ wcel 2098  Ⅎwnfc 2879   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  Vcvv 3473  ⦋csb 3894  ∅c0 4326  ∩ ciin 5001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-nul 4327  df-iin 5003

This theorem is referenced by:  eliin2  44513