Theorem eliin2f 45850

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 4960	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
2	1	adantl 486	. 2 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ∈ V) → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3		prcnel 3479	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
4	3	adantl 486	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
5		n0 4306	. . . . . . . 8 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
6	5	birani 508	. . . . . . 7 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 𝑦 ∈ 𝐵)
7		prcnel 3479	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
8	7	a1d 26	. . . . . . . . . 10 ⊢ (¬ 𝐴 ∈ V → (𝑦 ∈ 𝐵 → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
9	8	adantl 486	. . . . . . . . 9 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦 ∈ 𝐵 → ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
10	9	ancld 559	. . . . . . . 8 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝑦 ∈ 𝐵 → (𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)))
11	10	eximdv 1946	. . . . . . 7 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (∃𝑦 𝑦 ∈ 𝐵 → ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)))
12	6, 11	mpd 16	. . . . . 6 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
13		df-rex 3089	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
14	12, 13	sylibr 237	. . . . 5 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
15		eliin2f.1	. . . . . 6 ⊢ Ⅎ𝑥𝐵
16		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑦𝐵
17		nfv 1943	. . . . . 6 ⊢ Ⅎ𝑦 ¬ 𝐴 ∈ 𝐶
18		nfcsb1v 3876	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
19	18	nfel2 2942	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶
20	19	nfn 1886	. . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶
21		csbeq1a 3866	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
22	21	eleq2d 2848	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
23	22	notbid 321	. . . . . 6 ⊢ (𝑥 = 𝑦 → (¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶))
24	15, 16, 17, 20, 23	cbvrexfw 3305	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 ¬ 𝐴 ∈ ⦋𝑦 / 𝑥⦌𝐶)
25	14, 24	sylibr 237	. . . 4 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶)
26		rexnal 3116	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 ¬ 𝐴 ∈ 𝐶 ↔ ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
27	25, 26	sylib 221	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → ¬ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
28	4, 27	2falsed 379	. 2 ⊢ ((𝐵 ≠ ∅ ∧ ¬ 𝐴 ∈ V) → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
29	2, 28	pm2.61dan 824	1 ⊢ (𝐵 ≠ ∅ → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1808   ∈ wcel 2142  Ⅎwnfc 2909   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  Vcvv 3454  ⦋csb 3852  ∅c0 4285  ∩ ciin 4956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-nul 4286  df-iin 4958

This theorem is used by:  eliin2  45862