Metamath Proof Explorer


Theorem cvmsdisj

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

Ref Expression
Hypothesis cvmcov.1 S = k J s 𝒫 C | s = F -1 k u s v s u u v = F u C 𝑡 u Homeo J 𝑡 k
Assertion cvmsdisj T S U A T B T A = B A B =

Proof

Step Hyp Ref Expression
1 cvmcov.1 S = k J s 𝒫 C | s = F -1 k u s v s u u v = F u C 𝑡 u Homeo J 𝑡 k
2 df-ne A B ¬ A = B
3 1 cvmsi T S U U J T C T T = F -1 U u T v T u u v = F u C 𝑡 u Homeo J 𝑡 U
4 3 simp3d T S U T = F -1 U u T v T u u v = F u C 𝑡 u Homeo J 𝑡 U
5 4 simprd T S U u T v T u u v = F u C 𝑡 u Homeo J 𝑡 U
6 simpl v T u u v = F u C 𝑡 u Homeo J 𝑡 U v T u u v =
7 6 ralimi 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 T S U u T v T u u v =
9 sneq u = A u = A
10 9 difeq2d u = A T u = T A
11 ineq1 u = A u v = A v
12 11 eqeq1d u = A u v = A v =
13 10 12 raleqbidv u = A v T u u v = v T A A v =
14 13 rspccva u T v T u u v = A T v T A A v =
15 8 14 sylan T S U A T v T A A v =
16 necom A B B A
17 eldifsn B T A B T B A
18 17 biimpri B T B A B T A
19 16 18 sylan2b B T A B B T A
20 ineq2 v = B A v = A B
21 20 eqeq1d v = B A v = A B =
22 21 rspccv v T A A v = B T A A B =
23 15 19 22 syl2im T S U A T B T A B A B =
24 23 expd T S U A T B T A B A B =
25 24 3impia T S U A T B T A B A B =
26 2 25 syl5bir T S U A T B T ¬ A = B A B =
27 26 orrd T S U A T B T A = B A B =