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Theorem cvmsdisj 35275
Description: An even covering of 𝑈 is a disjoint union. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypothesis
Ref Expression
cvmcov.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
Assertion
Ref Expression
cvmsdisj ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
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
StepHypRef Expression
1 df-ne 2941 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 cvmcov.1 . . . . . . . . . . 11 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
32cvmsi 35270 . . . . . . . . . 10 (𝑇 ∈ (𝑆𝑈) → (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
43simp3d 1145 . . . . . . . . 9 (𝑇 ∈ (𝑆𝑈) → ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
54simprd 495 . . . . . . . 8 (𝑇 ∈ (𝑆𝑈) → ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))
6 simpl 482 . . . . . . . . 9 ((∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) → ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅)
76ralimi 3083 . . . . . . . 8 (∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) → ∀𝑢𝑇𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅)
85, 7syl 17 . . . . . . 7 (𝑇 ∈ (𝑆𝑈) → ∀𝑢𝑇𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅)
9 sneq 4636 . . . . . . . . . 10 (𝑢 = 𝐴 → {𝑢} = {𝐴})
109difeq2d 4126 . . . . . . . . 9 (𝑢 = 𝐴 → (𝑇 ∖ {𝑢}) = (𝑇 ∖ {𝐴}))
11 ineq1 4213 . . . . . . . . . 10 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
1211eqeq1d 2739 . . . . . . . . 9 (𝑢 = 𝐴 → ((𝑢𝑣) = ∅ ↔ (𝐴𝑣) = ∅))
1310, 12raleqbidv 3346 . . . . . . . 8 (𝑢 = 𝐴 → (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅))
1413rspccva 3621 . . . . . . 7 ((∀𝑢𝑇𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ 𝐴𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅)
158, 14sylan 580 . . . . . 6 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅)
16 necom 2994 . . . . . . 7 (𝐴𝐵𝐵𝐴)
17 eldifsn 4786 . . . . . . . 8 (𝐵 ∈ (𝑇 ∖ {𝐴}) ↔ (𝐵𝑇𝐵𝐴))
1817biimpri 228 . . . . . . 7 ((𝐵𝑇𝐵𝐴) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
1916, 18sylan2b 594 . . . . . 6 ((𝐵𝑇𝐴𝐵) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
20 ineq2 4214 . . . . . . . 8 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
2120eqeq1d 2739 . . . . . . 7 (𝑣 = 𝐵 → ((𝐴𝑣) = ∅ ↔ (𝐴𝐵) = ∅))
2221rspccv 3619 . . . . . 6 (∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅ → (𝐵 ∈ (𝑇 ∖ {𝐴}) → (𝐴𝐵) = ∅))
2315, 19, 22syl2im 40 . . . . 5 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐵𝑇𝐴𝐵) → (𝐴𝐵) = ∅))
2423expd 415 . . . 4 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐵𝑇 → (𝐴𝐵 → (𝐴𝐵) = ∅)))
25243impia 1118 . . 3 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴𝐵 → (𝐴𝐵) = ∅))
261, 25biimtrrid 243 . 2 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (¬ 𝐴 = 𝐵 → (𝐴𝐵) = ∅))
2726orrd 864 1 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907  cmpt 5225  ccnv 5684  cres 5687  cima 5688  cfv 6561  (class class class)co 7431  t crest 17465  Homeochmeo 23761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434
This theorem is referenced by:  cvmscld  35278  cvmsss2  35279  cvmseu  35281
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