Theorem pw1disj 4168

Description: Two unit power classes are disjoint iff the classes themselves are disjoint. (Contributed by SF, 26-Jan-2015.)

Assertion

Ref	Expression
pw1disj	⊢ ((𝒫1A ∩ 𝒫1B) = ∅ ↔ (A ∩ B) = ∅)

Proof of Theorem pw1disj

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

Step	Hyp	Ref	Expression
1		disj 3592	. . . . . 6 ⊢ ((𝒫1A ∩ 𝒫1B) = ∅ ↔ ∀y ∈ 𝒫1 A ¬ y ∈ 𝒫1B)
2		eleq1 2413	. . . . . . . 8 ⊢ (y = {x} → (y ∈ 𝒫1B ↔ {x} ∈ 𝒫1B))
3	2	notbid 285	. . . . . . 7 ⊢ (y = {x} → (¬ y ∈ 𝒫1B ↔ ¬ {x} ∈ 𝒫1B))
4	3	rspccv 2953	. . . . . 6 ⊢ (∀y ∈ 𝒫1 A ¬ y ∈ 𝒫1B → ({x} ∈ 𝒫1A → ¬ {x} ∈ 𝒫1B))
5	1, 4	sylbi 187	. . . . 5 ⊢ ((𝒫1A ∩ 𝒫1B) = ∅ → ({x} ∈ 𝒫1A → ¬ {x} ∈ 𝒫1B))
6		snelpw1 4147	. . . . 5 ⊢ ({x} ∈ 𝒫1A ↔ x ∈ A)
7		snelpw1 4147	. . . . . 6 ⊢ ({x} ∈ 𝒫1B ↔ x ∈ B)
8	7	notbii 287	. . . . 5 ⊢ (¬ {x} ∈ 𝒫1B ↔ ¬ x ∈ B)
9	5, 6, 8	3imtr3g 260	. . . 4 ⊢ ((𝒫1A ∩ 𝒫1B) = ∅ → (x ∈ A → ¬ x ∈ B))
10	9	ralrimiv 2697	. . 3 ⊢ ((𝒫1A ∩ 𝒫1B) = ∅ → ∀x ∈ A ¬ x ∈ B)
11		disj 3592	. . 3 ⊢ ((A ∩ B) = ∅ ↔ ∀x ∈ A ¬ x ∈ B)
12	10, 11	sylibr 203	. 2 ⊢ ((𝒫1A ∩ 𝒫1B) = ∅ → (A ∩ B) = ∅)
13		elpw1 4145	. . . . 5 ⊢ (x ∈ 𝒫1A ↔ ∃y ∈ A x = {y})
14		disj 3592	. . . . . . . . 9 ⊢ ((A ∩ B) = ∅ ↔ ∀y ∈ A ¬ y ∈ B)
15		rsp 2675	. . . . . . . . 9 ⊢ (∀y ∈ A ¬ y ∈ B → (y ∈ A → ¬ y ∈ B))
16	14, 15	sylbi 187	. . . . . . . 8 ⊢ ((A ∩ B) = ∅ → (y ∈ A → ¬ y ∈ B))
17	16	imp 418	. . . . . . 7 ⊢ (((A ∩ B) = ∅ ∧ y ∈ A) → ¬ y ∈ B)
18		eleq1 2413	. . . . . . . . 9 ⊢ (x = {y} → (x ∈ 𝒫1B ↔ {y} ∈ 𝒫1B))
19		snelpw1 4147	. . . . . . . . 9 ⊢ ({y} ∈ 𝒫1B ↔ y ∈ B)
20	18, 19	syl6bb 252	. . . . . . . 8 ⊢ (x = {y} → (x ∈ 𝒫1B ↔ y ∈ B))
21	20	notbid 285	. . . . . . 7 ⊢ (x = {y} → (¬ x ∈ 𝒫1B ↔ ¬ y ∈ B))
22	17, 21	syl5ibrcom 213	. . . . . 6 ⊢ (((A ∩ B) = ∅ ∧ y ∈ A) → (x = {y} → ¬ x ∈ 𝒫1B))
23	22	rexlimdva 2739	. . . . 5 ⊢ ((A ∩ B) = ∅ → (∃y ∈ A x = {y} → ¬ x ∈ 𝒫1B))
24	13, 23	syl5bi 208	. . . 4 ⊢ ((A ∩ B) = ∅ → (x ∈ 𝒫1A → ¬ x ∈ 𝒫1B))
25	24	ralrimiv 2697	. . 3 ⊢ ((A ∩ B) = ∅ → ∀x ∈ 𝒫1 A ¬ x ∈ 𝒫1B)
26		disj 3592	. . 3 ⊢ ((𝒫1A ∩ 𝒫1B) = ∅ ↔ ∀x ∈ 𝒫1 A ¬ x ∈ 𝒫1B)
27	25, 26	sylibr 203	. 2 ⊢ ((A ∩ B) = ∅ → (𝒫1A ∩ 𝒫1B) = ∅)
28	12, 27	impbii 180	1 ⊢ ((𝒫1A ∩ 𝒫1B) = ∅ ↔ (A ∩ B) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616   ∩ cin 3209  ∅c0 3551  {csn 3738  𝒫₁cpw1 4136

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742  df-1c 4137  df-pw1 4138

This theorem is used by: (None)