Theorem disjif2 30344

Description: Property of a disjoint collection: if 𝐵(𝑥) and 𝐵(𝑌) = 𝐷 have a common element 𝑍, then 𝑥 = 𝑌. (Contributed by Thierry Arnoux, 6-Apr-2017.)

Hypotheses

Ref	Expression
disjif2.1	⊢ Ⅎ𝑥𝐴
disjif2.2	⊢ Ⅎ𝑥𝐶
disjif2.3	⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶)

Assertion

Ref	Expression
disjif2	⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶)) → 𝑥 = 𝑌)

Distinct variable group:   𝑥,𝑌

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

Proof of Theorem disjif2

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

Step	Hyp	Ref	Expression
1		inelcm 4372	. 2 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → (𝐵 ∩ 𝐶) ≠ ∅)
2		disjif2.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
3	2	disjorsf 30343	. . . . . . 7 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅))
4		equequ1 2032	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑧 ↔ 𝑥 = 𝑧))
5		csbeq1 3831	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵)
6		csbid 3841	. . . . . . . . . . . 12 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵
7	5, 6	eqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)
8	7	ineq1d 4138	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵))
9	8	eqeq1d 2800	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ ↔ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅))
10	4, 9	orbi12d 916	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) ↔ (𝑥 = 𝑧 ∨ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅)))
11		eqeq2 2810	. . . . . . . . 9 ⊢ (𝑧 = 𝑌 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑌))
12		nfcv 2955	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑌
13		disjif2.2	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐶
14		disjif2.3	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶)
15	12, 13, 14	csbhypf 3856	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑌 → ⦋𝑧 / 𝑥⦌𝐵 = 𝐶)
16	15	ineq2d 4139	. . . . . . . . . 10 ⊢ (𝑧 = 𝑌 → (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = (𝐵 ∩ 𝐶))
17	16	eqeq1d 2800	. . . . . . . . 9 ⊢ (𝑧 = 𝑌 → ((𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ ↔ (𝐵 ∩ 𝐶) = ∅))
18	11, 17	orbi12d 916	. . . . . . . 8 ⊢ (𝑧 = 𝑌 → ((𝑥 = 𝑧 ∨ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) ↔ (𝑥 = 𝑌 ∨ (𝐵 ∩ 𝐶) = ∅)))
19	10, 18	rspc2v 3581	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → (𝑥 = 𝑌 ∨ (𝐵 ∩ 𝐶) = ∅)))
20	3, 19	syl5bi 245	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (Disj 𝑥 ∈ 𝐴 𝐵 → (𝑥 = 𝑌 ∨ (𝐵 ∩ 𝐶) = ∅)))
21	20	impcom 411	. . . . 5 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑥 = 𝑌 ∨ (𝐵 ∩ 𝐶) = ∅))
22	21	ord 861	. . . 4 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (¬ 𝑥 = 𝑌 → (𝐵 ∩ 𝐶) = ∅))
23	22	necon1ad 3004	. . 3 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐵 ∩ 𝐶) ≠ ∅ → 𝑥 = 𝑌))
24	23	3impia 1114	. 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝐵 ∩ 𝐶) ≠ ∅) → 𝑥 = 𝑌)
25	1, 24	syl3an3 1162	1 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝑥 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) ∧ (𝑍 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶)) → 𝑥 = 𝑌)

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936   ≠ wne 2987  ∀wral 3106  ⦋csb 3828   ∩ cin 3880  ∅c0 4243  Disj wdisj 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-in 3888  df-nul 4244  df-disj 4996

This theorem is referenced by:  disjabrexf  30346