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Theorem cvmsdisj 35254
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 2938 . . 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 35249 . . . . . . . . . 10 (𝑇 ∈ (𝑆𝑈) → (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
43simp3d 1143 . . . . . . . . 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 3080 . . . . . . . 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 4640 . . . . . . . . . 10 (𝑢 = 𝐴 → {𝑢} = {𝐴})
109difeq2d 4135 . . . . . . . . 9 (𝑢 = 𝐴 → (𝑇 ∖ {𝑢}) = (𝑇 ∖ {𝐴}))
11 ineq1 4220 . . . . . . . . . 10 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
1211eqeq1d 2736 . . . . . . . . 9 (𝑢 = 𝐴 → ((𝑢𝑣) = ∅ ↔ (𝐴𝑣) = ∅))
1310, 12raleqbidv 3343 . . . . . . . 8 (𝑢 = 𝐴 → (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅))
1413rspccva 3620 . . . . . . 7 ((∀𝑢𝑇𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ 𝐴𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅)
158, 14sylan 580 . . . . . 6 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ∀𝑣 ∈ (𝑇 ∖ {𝐴})(𝐴𝑣) = ∅)
16 necom 2991 . . . . . . 7 (𝐴𝐵𝐵𝐴)
17 eldifsn 4790 . . . . . . . 8 (𝐵 ∈ (𝑇 ∖ {𝐴}) ↔ (𝐵𝑇𝐵𝐴))
1817biimpri 228 . . . . . . 7 ((𝐵𝑇𝐵𝐴) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
1916, 18sylan2b 594 . . . . . 6 ((𝐵𝑇𝐴𝐵) → 𝐵 ∈ (𝑇 ∖ {𝐴}))
20 ineq2 4221 . . . . . . . 8 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
2120eqeq1d 2736 . . . . . . 7 (𝑣 = 𝐵 → ((𝐴𝑣) = ∅ ↔ (𝐴𝐵) = ∅))
2221rspccv 3618 . . . . . 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 1116 . . 3 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴𝐵 → (𝐴𝐵) = ∅))
261, 25biimtrrid 243 . 2 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (¬ 𝐴 = 𝐵 → (𝐴𝐵) = ∅))
2726orrd 863 1 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝐵𝑇) → (𝐴 = 𝐵 ∨ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  {crab 3432  cdif 3959  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604  {csn 4630   cuni 4911  cmpt 5230  ccnv 5687  cres 5690  cima 5691  cfv 6562  (class class class)co 7430  t crest 17466  Homeochmeo 23776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433
This theorem is referenced by:  cvmscld  35257  cvmsss2  35258  cvmseu  35260
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