Theorem disji2 3930

Description: Property of a disjoint collection: if 𝐵(𝑋) = 𝐶 and 𝐵(𝑌) = 𝐷, and 𝑋 ≠ 𝑌, then 𝐶 and 𝐷 are disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
disji.1	⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
disji.2	⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)

Assertion

Ref	Expression
disji2	⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ 𝑋 ≠ 𝑌) → (𝐶 ∩ 𝐷) = ∅)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑋   𝑥,𝑌

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disji2

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjnims 3929	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 ≠ 𝑧 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅))
2		neeq1 2322	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≠ 𝑧 ↔ 𝑋 ≠ 𝑧))
3		nfcv 2282	. . . . . . . 8 ⊢ Ⅎ𝑥𝑋
4		nfcv 2282	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
5		disji.1	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
6	3, 4, 5	csbhypf 3043	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
7	6	ineq1d 3281	. . . . . 6 ⊢ (𝑦 = 𝑋 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = (𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵))
8	7	eqeq1d 2149	. . . . 5 ⊢ (𝑦 = 𝑋 → ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ ↔ (𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅))
9	2, 8	imbi12d 233	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 ≠ 𝑧 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) ↔ (𝑋 ≠ 𝑧 → (𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅)))
10		neeq2 2323	. . . . 5 ⊢ (𝑧 = 𝑌 → (𝑋 ≠ 𝑧 ↔ 𝑋 ≠ 𝑌))
11		nfcv 2282	. . . . . . . 8 ⊢ Ⅎ𝑥𝑌
12		nfcv 2282	. . . . . . . 8 ⊢ Ⅎ𝑥𝐷
13		disji.2	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
14	11, 12, 13	csbhypf 3043	. . . . . . 7 ⊢ (𝑧 = 𝑌 → ⦋𝑧 / 𝑥⦌𝐵 = 𝐷)
15	14	ineq2d 3282	. . . . . 6 ⊢ (𝑧 = 𝑌 → (𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵) = (𝐶 ∩ 𝐷))
16	15	eqeq1d 2149	. . . . 5 ⊢ (𝑧 = 𝑌 → ((𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ ↔ (𝐶 ∩ 𝐷) = ∅))
17	10, 16	imbi12d 233	. . . 4 ⊢ (𝑧 = 𝑌 → ((𝑋 ≠ 𝑧 → (𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) ↔ (𝑋 ≠ 𝑌 → (𝐶 ∩ 𝐷) = ∅)))
18	9, 17	rspc2v 2806	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 ≠ 𝑧 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → (𝑋 ≠ 𝑌 → (𝐶 ∩ 𝐷) = ∅)))
19	1, 18	mpan9 279	. 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋 ≠ 𝑌 → (𝐶 ∩ 𝐷) = ∅))
20	19	3impia 1179	1 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ 𝑋 ≠ 𝑌) → (𝐶 ∩ 𝐷) = ∅)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 1481   ≠ wne 2309  ∀wral 2417  ⦋csb 3007   ∩ cin 3075  ∅c0 3368  Disj wdisj 3914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-in 3082  df-nul 3369  df-disj 3915

This theorem is referenced by: (None)