Theorem disji2 4026

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 4025	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 ≠ 𝑧 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅))
2		neeq1 2380	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≠ 𝑧 ↔ 𝑋 ≠ 𝑧))
3		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝑋
4		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
5		disji.1	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
6	3, 4, 5	csbhypf 3123	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶)
7	6	ineq1d 3363	. . . . . 6 ⊢ (𝑦 = 𝑋 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = (𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵))
8	7	eqeq1d 2205	. . . . 5 ⊢ (𝑦 = 𝑋 → ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ ↔ (𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅))
9	2, 8	imbi12d 234	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 ≠ 𝑧 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) ↔ (𝑋 ≠ 𝑧 → (𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅)))
10		neeq2 2381	. . . . 5 ⊢ (𝑧 = 𝑌 → (𝑋 ≠ 𝑧 ↔ 𝑋 ≠ 𝑌))
11		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝑌
12		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑥𝐷
13		disji.2	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
14	11, 12, 13	csbhypf 3123	. . . . . . 7 ⊢ (𝑧 = 𝑌 → ⦋𝑧 / 𝑥⦌𝐵 = 𝐷)
15	14	ineq2d 3364	. . . . . 6 ⊢ (𝑧 = 𝑌 → (𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵) = (𝐶 ∩ 𝐷))
16	15	eqeq1d 2205	. . . . 5 ⊢ (𝑧 = 𝑌 → ((𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ ↔ (𝐶 ∩ 𝐷) = ∅))
17	10, 16	imbi12d 234	. . . 4 ⊢ (𝑧 = 𝑌 → ((𝑋 ≠ 𝑧 → (𝐶 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) ↔ (𝑋 ≠ 𝑌 → (𝐶 ∩ 𝐷) = ∅)))
18	9, 17	rspc2v 2881	. . 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 281	. 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋 ≠ 𝑌 → (𝐶 ∩ 𝐷) = ∅))
20	19	3impia 1202	1 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ 𝑋 ≠ 𝑌) → (𝐶 ∩ 𝐷) = ∅)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  ⦋csb 3084   ∩ cin 3156  ∅c0 3450  Disj wdisj 4010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-in 3163  df-nul 3451  df-disj 4011

This theorem is referenced by: (None)