Theorem disjiun 3924

Description: A disjoint collection yields disjoint indexed unions for disjoint index sets. (Contributed by Mario Carneiro, 26-Mar-2015.) (Revised by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjiun	⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵) = ∅)

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

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjiun

Dummy variables 𝑢 𝑣 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-disj 3907	. . . 4 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		elin 3259	. . . . . . . . . 10 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵) ↔ (𝑦 ∈ ∪ 𝑥 ∈ 𝐶 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐷 𝐵))
3		eliun 3817	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐶 𝐵 ↔ ∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵)
4		eliun 3817	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐷 𝐵 ↔ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)
5	3, 4	anbi12i 455	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐶 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐷 𝐵) ↔ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵))
6	2, 5	bitri 183	. . . . . . . . 9 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵) ↔ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵))
7		nfv 1508	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵
8	7	rmo3 3000	. . . . . . . . . . 11 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤))
9		simprl 520	. . . . . . . . . . . . . 14 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) → ∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵)
10		nfv 1508	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑢 𝑦 ∈ 𝐵
11		nfcsb1v 3035	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
12	11	nfcri 2275	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵
13		csbeq1a 3012	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
14	13	eleq2d 2209	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵))
15	10, 12, 14	cbvrex 2651	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ↔ ∃𝑢 ∈ 𝐶 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)
16	9, 15	sylib 121	. . . . . . . . . . . . 13 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) → ∃𝑢 ∈ 𝐶 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)
17		simplrr 525	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) → ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)
18		nfv 1508	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑣 𝑦 ∈ 𝐵
19		nfcsb1v 3035	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑣 / 𝑥⦌𝐵
20	19	nfcri 2275	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵
21		csbeq1a 3012	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑣 → 𝐵 = ⦋𝑣 / 𝑥⦌𝐵)
22	21	eleq2d 2209	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑣 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵))
23	18, 20, 22	cbvrex 2651	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵 ↔ ∃𝑣 ∈ 𝐷 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)
24	17, 23	sylib 121	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) → ∃𝑣 ∈ 𝐷 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)
25		simplrl 524	. . . . . . . . . . . . . . 15 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑢 ∈ 𝐶)
26		simprl 520	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → 𝐶 ⊆ 𝐴)
27	26	ad3antrrr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝐶 ⊆ 𝐴)
28	27, 25	sseldd 3098	. . . . . . . . . . . . . . . . . 18 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑢 ∈ 𝐴)
29		simprr 521	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → 𝐷 ⊆ 𝐴)
30	29	ad3antrrr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝐷 ⊆ 𝐴)
31		simprl 520	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑣 ∈ 𝐷)
32	30, 31	sseldd 3098	. . . . . . . . . . . . . . . . . 18 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑣 ∈ 𝐴)
33	28, 32	jca 304	. . . . . . . . . . . . . . . . 17 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴))
34		simp-4l 530	. . . . . . . . . . . . . . . . 17 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → ∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤))
35		simplrr 525	. . . . . . . . . . . . . . . . . 18 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)
36		simprr 521	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)
37	20, 22	sbie 1764	. . . . . . . . . . . . . . . . . . 19 ⊢ ([𝑣 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)
38	36, 37	sylibr 133	. . . . . . . . . . . . . . . . . 18 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → [𝑣 / 𝑥]𝑦 ∈ 𝐵)
39	35, 38	jca 304	. . . . . . . . . . . . . . . . 17 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → (𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑣 / 𝑥]𝑦 ∈ 𝐵))
40		nfs1v 1912	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑦 ∈ 𝐵
41	12, 40	nfan 1544	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥(𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵)
42		nfv 1508	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥 𝑢 = 𝑤
43	41, 42	nfim 1551	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑤)
44		nfv 1508	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑤((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑣 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑣)
45	14	anbi1d 460	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑢 → ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵)))
46		equequ1 1688	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑤 ↔ 𝑢 = 𝑤))
47	45, 46	imbi12d 233	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑢 → (((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ↔ ((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑤)))
48		sbequ 1812	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑣 → ([𝑤 / 𝑥]𝑦 ∈ 𝐵 ↔ [𝑣 / 𝑥]𝑦 ∈ 𝐵))
49	48	anbi2d 459	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑣 → ((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) ↔ (𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑣 / 𝑥]𝑦 ∈ 𝐵)))
50		equequ2 1689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑣 → (𝑢 = 𝑤 ↔ 𝑢 = 𝑣))
51	49, 50	imbi12d 233	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑣 → (((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑤) ↔ ((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑣 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑣)))
52	43, 44, 47, 51	rspc2 2800	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) → ((𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵 ∧ [𝑣 / 𝑥]𝑦 ∈ 𝐵) → 𝑢 = 𝑣)))
53	33, 34, 39, 52	syl3c 63	. . . . . . . . . . . . . . . 16 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑢 = 𝑣)
54	53, 31	eqeltrd 2216	. . . . . . . . . . . . . . 15 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → 𝑢 ∈ 𝐷)
55		inelcm 3423	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ 𝐶 ∧ 𝑢 ∈ 𝐷) → (𝐶 ∩ 𝐷) ≠ ∅)
56	25, 54, 55	syl2anc 408	. . . . . . . . . . . . . 14 ⊢ (((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) ∧ (𝑣 ∈ 𝐷 ∧ 𝑦 ∈ ⦋𝑣 / 𝑥⦌𝐵)) → (𝐶 ∩ 𝐷) ≠ ∅)
57	24, 56	rexlimddv 2554	. . . . . . . . . . . . 13 ⊢ ((((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) ∧ (𝑢 ∈ 𝐶 ∧ 𝑦 ∈ ⦋𝑢 / 𝑥⦌𝐵)) → (𝐶 ∩ 𝐷) ≠ ∅)
58	16, 57	rexlimddv 2554	. . . . . . . . . . . 12 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) ∧ (∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵)) → (𝐶 ∩ 𝐷) ≠ ∅)
59	58	exp31 361	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝑦 ∈ 𝐵 ∧ [𝑤 / 𝑥]𝑦 ∈ 𝐵) → 𝑥 = 𝑤) → ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) → ((∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵) → (𝐶 ∩ 𝐷) ≠ ∅)))
60	8, 59	sylbi 120	. . . . . . . . . 10 ⊢ (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) → ((∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵) → (𝐶 ∩ 𝐷) ≠ ∅)))
61	60	impcom 124	. . . . . . . . 9 ⊢ (((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) ∧ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ((∃𝑥 ∈ 𝐶 𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐷 𝑦 ∈ 𝐵) → (𝐶 ∩ 𝐷) ≠ ∅))
62	6, 61	syl5bi 151	. . . . . . . 8 ⊢ (((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) ∧ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → (𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵) → (𝐶 ∩ 𝐷) ≠ ∅))
63	62	necon2bd 2366	. . . . . . 7 ⊢ (((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) ∧ ∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ((𝐶 ∩ 𝐷) = ∅ → ¬ 𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵)))
64	63	impancom 258	. . . . . 6 ⊢ (((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ¬ 𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵)))
65	64	3impa 1176	. . . . 5 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅) → (∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ¬ 𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵)))
66	65	alimdv 1851	. . . 4 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅) → (∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∀𝑦 ¬ 𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵)))
67	1, 66	syl5bi 151	. . 3 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅) → (Disj 𝑥 ∈ 𝐴 𝐵 → ∀𝑦 ¬ 𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵)))
68	67	impcom 124	. 2 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅)) → ∀𝑦 ¬ 𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵))
69		eq0 3381	. 2 ⊢ ((∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵) = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵))
70	68, 69	sylibr 133	1 ⊢ ((Disj 𝑥 ∈ 𝐴 𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴 ∧ (𝐶 ∩ 𝐷) = ∅)) → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ 𝐷 𝐵) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∧ w3a 962  ∀wal 1329   = wceq 1331   ∈ wcel 1480  [wsb 1735   ≠ wne 2308  ∀wral 2416  ∃wrex 2417  ∃*wrmo 2419  ⦋csb 3003   ∩ cin 3070   ⊆ wss 3071  ∅c0 3363  ∪ ciun 3813  Disj wdisj 3906

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rmo 2424  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-iun 3815  df-disj 3907

This theorem is referenced by:  iunfidisj  6834  fsumiun  11246