Theorem eliin2f 41368

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 4923	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
2	1	adantl 484	. 2 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ∈ V) → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3		prcnel 3518	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
4	3	adantl 484	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
5		n0 4309	. . . . . . . . 9 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
6	5	biimpi 218	. . . . . . . 8 ⊢ (𝐵 ≠ ∅ → ∃𝑦 𝑦 ∈ 𝐵)
7	6	adantr 483	. . . . . . 7 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 𝑦 ∈ 𝐵)
8		prcnel 3518	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
9	8	a1d 25	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ V → (𝑦 ∈ 𝐵 → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
10	9	adantl 484	. . . . . . . . 9 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦 ∈ 𝐵 → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
11	10	ancld 553	. . . . . . . 8 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)))
12	11	eximdv 1914	. . . . . . 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 3144	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
15	13, 14	sylibr 236	. . . . 5 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
16		eliin2f.1	. . . . . 6 ⊢ Ⅎ𝑥𝐵
17		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑦𝐵
18		nfv 1911	. . . . . 6 ⊢ Ⅎ𝑦 ¬ 𝐴 ∈ 𝐶
19		nfcsb1v 3906	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
20	19	nfel2 2996	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶
21	20	nfn 1853	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶
22		csbeq1a 3896	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
23	22	eleq2d 2898	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
24	23	notbid 320	. . . . . 6 ⊢ (𝑥 = 𝑦 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
25	16, 17, 18, 21, 24	cbvrexfw 3438	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
26	15, 25	sylibr 236	. . . 4 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶)
27		rexnal 3238	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶 ↔ ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
28	26, 27	sylib 220	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
29	4, 28	2falsed 379	. 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 208   ∧ wa 398  ∃wex 1776   ∈ wcel 2110  Ⅎwnfc 2961   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494  ⦋csb 3882  ∅c0 4290  ∩ ciin 4919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-nul 4291  df-iin 4921

This theorem is referenced by:  eliin2  41380