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Theorem ddeval0 33982
Description: Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)
Assertion
Ref Expression
ddeval0 ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0)

Proof of Theorem ddeval0
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 reex 11231 . . . . 5 ℝ ∈ V
21ssex 5322 . . . 4 (𝐴 ⊆ ℝ → 𝐴 ∈ V)
3 elpwg 4607 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ))
43biimpar 476 . . . 4 ((𝐴 ∈ V ∧ 𝐴 ⊆ ℝ) → 𝐴 ∈ 𝒫 ℝ)
52, 4mpancom 686 . . 3 (𝐴 ⊆ ℝ → 𝐴 ∈ 𝒫 ℝ)
6 eleq2 2814 . . . . 5 (𝑎 = 𝐴 → (0 ∈ 𝑎 ↔ 0 ∈ 𝐴))
76ifbid 4553 . . . 4 (𝑎 = 𝐴 → if(0 ∈ 𝑎, 1, 0) = if(0 ∈ 𝐴, 1, 0))
8 df-dde 33980 . . . 4 δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈ 𝑎, 1, 0))
9 1ex 11242 . . . . 5 1 ∈ V
10 c0ex 11240 . . . . 5 0 ∈ V
119, 10ifex 4580 . . . 4 if(0 ∈ 𝐴, 1, 0) ∈ V
127, 8, 11fvmpt 7004 . . 3 (𝐴 ∈ 𝒫 ℝ → (δ‘𝐴) = if(0 ∈ 𝐴, 1, 0))
135, 12syl 17 . 2 (𝐴 ⊆ ℝ → (δ‘𝐴) = if(0 ∈ 𝐴, 1, 0))
14 iffalse 4539 . 2 (¬ 0 ∈ 𝐴 → if(0 ∈ 𝐴, 1, 0) = 0)
1513, 14sylan9eq 2785 1 ((𝐴 ⊆ ℝ ∧ ¬ 0 ∈ 𝐴) → (δ‘𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  wss 3944  ifcif 4530  𝒫 cpw 4604  cfv 6549  cr 11139  0cc0 11140  1c1 11141  δcdde 33979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-mulcl 11202  ax-i2m1 11208
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-dde 33980
This theorem is referenced by:  ddemeas  33983
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