Theorem disjiund 5142

Description: Conditions for a collection of index unions of sets 𝐴(𝑎, 𝑏) for 𝑎 ∈ 𝑉 and 𝑏 ∈ 𝑊 to be disjoint. (Contributed by AV, 9-Jan-2022.)

Hypotheses

Ref	Expression
disjiund.1	⊢ (𝑎 = 𝑐 → 𝐴 = 𝐶)
disjiund.2	⊢ (𝑏 = 𝑑 → 𝐶 = 𝐷)
disjiund.3	⊢ (𝑎 = 𝑐 → 𝑊 = 𝑋)
disjiund.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐷) → 𝑎 = 𝑐)

Assertion

Ref	Expression
disjiund	⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ 𝑊 𝐴)

Distinct variable groups:   𝐴,𝑐,𝑑,𝑥   𝐶,𝑎,𝑑,𝑥   𝐷,𝑏   𝑉,𝑎,𝑐   𝑊,𝑏,𝑐,𝑑,𝑥   𝑋,𝑎,𝑏,𝑑,𝑥   𝜑,𝑎,𝑏,𝑐,𝑑,𝑥

Allowed substitution hints:   𝐴(𝑎,𝑏)   𝐶(𝑏,𝑐)   𝐷(𝑥,𝑎,𝑐,𝑑)   𝑉(𝑥,𝑏,𝑑)   𝑊(𝑎)   𝑋(𝑐)

Proof of Theorem disjiund

Step	Hyp	Ref	Expression
1		eliun 5004	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑏 ∈ 𝑊 𝐴 ↔ ∃𝑏 ∈ 𝑊 𝑥 ∈ 𝐴)
2		eliun 5004	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑏 ∈ 𝑋 𝐶 ↔ ∃𝑏 ∈ 𝑋 𝑥 ∈ 𝐶)
3		disjiund.2	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑑 → 𝐶 = 𝐷)
4	3	eleq2d 2815	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ 𝐷))
5	4	cbvrexvw 3233	. . . . . . . . . . . . 13 ⊢ (∃𝑏 ∈ 𝑋 𝑥 ∈ 𝐶 ↔ ∃𝑑 ∈ 𝑋 𝑥 ∈ 𝐷)
6		disjiund.4	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐷) → 𝑎 = 𝑐)
7	6	3exp 1116	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐷 → 𝑎 = 𝑐)))
8	7	rexlimdvw 3157	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∃𝑏 ∈ 𝑊 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐷 → 𝑎 = 𝑐)))
9	8	imp 405	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∃𝑏 ∈ 𝑊 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐷 → 𝑎 = 𝑐))
10	9	rexlimdvw 3157	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∃𝑏 ∈ 𝑊 𝑥 ∈ 𝐴) → (∃𝑑 ∈ 𝑋 𝑥 ∈ 𝐷 → 𝑎 = 𝑐))
11	5, 10	biimtrid 241	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∃𝑏 ∈ 𝑊 𝑥 ∈ 𝐴) → (∃𝑏 ∈ 𝑋 𝑥 ∈ 𝐶 → 𝑎 = 𝑐))
12	2, 11	biimtrid 241	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∃𝑏 ∈ 𝑊 𝑥 ∈ 𝐴) → (𝑥 ∈ ∪ 𝑏 ∈ 𝑋 𝐶 → 𝑎 = 𝑐))
13	12	con3d 152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∃𝑏 ∈ 𝑊 𝑥 ∈ 𝐴) → (¬ 𝑎 = 𝑐 → ¬ 𝑥 ∈ ∪ 𝑏 ∈ 𝑋 𝐶))
14	13	impancom 450	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑐) → (∃𝑏 ∈ 𝑊 𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ ∪ 𝑏 ∈ 𝑋 𝐶))
15	1, 14	biimtrid 241	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑐) → (𝑥 ∈ ∪ 𝑏 ∈ 𝑊 𝐴 → ¬ 𝑥 ∈ ∪ 𝑏 ∈ 𝑋 𝐶))
16	15	ralrimiv 3142	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑐) → ∀𝑥 ∈ ∪ 𝑏 ∈ 𝑊 𝐴 ¬ 𝑥 ∈ ∪ 𝑏 ∈ 𝑋 𝐶)
17		disj 4451	. . . . . . 7 ⊢ ((∪ 𝑏 ∈ 𝑊 𝐴 ∩ ∪ 𝑏 ∈ 𝑋 𝐶) = ∅ ↔ ∀𝑥 ∈ ∪ 𝑏 ∈ 𝑊 𝐴 ¬ 𝑥 ∈ ∪ 𝑏 ∈ 𝑋 𝐶)
18	16, 17	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑎 = 𝑐) → (∪ 𝑏 ∈ 𝑊 𝐴 ∩ ∪ 𝑏 ∈ 𝑋 𝐶) = ∅)
19	18	ex 411	. . . . 5 ⊢ (𝜑 → (¬ 𝑎 = 𝑐 → (∪ 𝑏 ∈ 𝑊 𝐴 ∩ ∪ 𝑏 ∈ 𝑋 𝐶) = ∅))
20	19	orrd 861	. . . 4 ⊢ (𝜑 → (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ 𝑊 𝐴 ∩ ∪ 𝑏 ∈ 𝑋 𝐶) = ∅))
21	20	a1d 25	. . 3 ⊢ (𝜑 → ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ 𝑊 𝐴 ∩ ∪ 𝑏 ∈ 𝑋 𝐶) = ∅)))
22	21	ralrimivv 3196	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ 𝑉 (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ 𝑊 𝐴 ∩ ∪ 𝑏 ∈ 𝑋 𝐶) = ∅))
23		disjiund.3	. . 3 ⊢ (𝑎 = 𝑐 → 𝑊 = 𝑋)
24		disjiund.1	. . 3 ⊢ (𝑎 = 𝑐 → 𝐴 = 𝐶)
25	23, 24	disjiunb 5141	. 2 ⊢ (Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ 𝑊 𝐴 ↔ ∀𝑎 ∈ 𝑉 ∀𝑐 ∈ 𝑉 (𝑎 = 𝑐 ∨ (∪ 𝑏 ∈ 𝑊 𝐴 ∩ ∪ 𝑏 ∈ 𝑋 𝐶) = ∅))
26	22, 25	sylibr 233	1 ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ 𝑊 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067   ∩ cin 3948  ∅c0 4326  ∪ ciun 5000  Disj wdisj 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2529  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rmo 3374  df-v 3475  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4327  df-iun 5002  df-disj 5118

This theorem is referenced by:  2wspiundisj  29794