Theorem cvmsdisj 36004

Description: An even covering of 𝑈 is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypothesis

Ref	Expression
cvmcov.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})

Assertion

Ref	Expression
cvmsdisj	⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 = 𝐵 ∨ (𝐴 ∩ 𝐵) = ∅))

Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣   𝑘,𝐽,𝑠,𝑢,𝑣   𝑈,𝑘,𝑠,𝑢,𝑣   𝑇,𝑠,𝑢,𝑣   𝑢,𝐴,𝑣   𝑣,𝐵

Allowed substitution hints:   𝐴(𝑘, 𝑠)   𝐵(𝑢, 𝑘, 𝑠)   𝑆(𝑣, 𝑢, 𝑘, 𝑠)   𝑇(𝑘)

Proof of Theorem cvmsdisj

Step	Hyp	Ref	Expression
1		df-ne 2957	. . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
2		cvmcov.1	. . . . . . . . . . 11 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
3	2	cvmsi 35999	. . . . . . . . . 10 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))
4	3	simp3d 1162	. . . . . . . . 9 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
5	4	simprd 501	. . . . . . . 8 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))
6		simpl 488	. . . . . . . . 9 ⊢ ((∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))) → ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅)
7	6	ralimi 3100	. . . . . . . 8 ⊢ (∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))) → ∀𝑢 ∈ 𝑇 ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅)
8	5, 7	syl 18	. . . . . . 7 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∀𝑢 ∈ 𝑇 ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅)
9		sneq 4594	. . . . . . . . . 10 ⊢ (𝑢 = 𝐴 → {𝑢} = {𝐴})
10	9	difeq2d 4074	. . . . . . . . 9 ⊢ (𝑢 = 𝐴 → (𝑇 ∖ {𝑢}) = (𝑇 ∖ {𝐴}))
11		ineq1 4159	. . . . . . . . . 10 ⊢ (𝑢 = 𝐴 → (𝑢 ∩ 𝑣) = (𝐴 ∩ 𝑣))
12	11	eqeq1d 2763	. . . . . . . . 9 ⊢ (𝑢 = 𝐴 → ((𝑢 ∩ 𝑣) = ∅ ↔ (𝐴 ∩ 𝑣) = ∅))
13	10, 12	raleqbidv 3335	. . . . . . . 8 ⊢ (𝑢 = 𝐴 → (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑣) = ∅))
14	13	rspccva 3576	. . . . . . 7 ⊢ ((∀𝑢 ∈ 𝑇 ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ 𝐴 ∈ 𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑣) = ∅)
15	8, 14	sylan 592	. . . . . 6 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑣) = ∅)
16		necom 3009	. . . . . . 7 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
17		eldifsn 4748	. . . . . . . 8 ⊢ (𝐵 ∈ (𝑇 ∖ {𝐴}) ↔ (𝐵 ∈ 𝑇 ∧ 𝐵 ≠ 𝐴))
18	17	biimpri 231	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑇 ∧ 𝐵 ≠ 𝐴) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
19	16, 18	sylan2b 606	. . . . . 6 ⊢ ((𝐵 ∈ 𝑇 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
20		ineq2 4160	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝐴 ∩ 𝑣) = (𝐴 ∩ 𝐵))
21	20	eqeq1d 2763	. . . . . . 7 ⊢ (𝑣 = 𝐵 → ((𝐴 ∩ 𝑣) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
22	21	rspccv 3574	. . . . . 6 ⊢ (∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑣) = ∅ → (𝐵 ∈ (𝑇 ∖ {𝐴}) → (𝐴 ∩ 𝐵) = ∅))
23	15, 19, 22	syl2im 41	. . . . 5 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((𝐵 ∈ 𝑇 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∩ 𝐵) = ∅))
24	23	expd 421	. . . 4 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐵 ∈ 𝑇 → (𝐴 ≠ 𝐵 → (𝐴 ∩ 𝐵) = ∅)))
25	24	3impia 1135	. . 3 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 ≠ 𝐵 → (𝐴 ∩ 𝐵) = ∅))
26	1, 25	biimtrrid 246	. 2 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (¬ 𝐴 = 𝐵 → (𝐴 ∩ 𝐵) = ∅))
27	26	orrd 877	1 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑇) → (𝐴 = 𝐵 ∨ (𝐴 ∩ 𝐵) = ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867   ↦ cmpt 5186  ^◡ccnv 5650   ↾ cres 5653   “ cima 5654  ‘cfv 6531  (class class class)co 7412   ↾_t crest 17571  Homeochmeo 24052

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415

This theorem is used by:  cvmscld  36007  cvmsss2  36008  cvmseu  36010