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Theorem ditgex 25369
Description: A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.)
Assertion
Ref Expression
ditgex ⨜[𝐴𝐵]𝐶 d𝑥 ∈ V

Proof of Theorem ditgex
StepHypRef Expression
1 df-ditg 25364 . 2 ⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
2 itgex 25288 . . 3 ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ V
3 negex 11458 . . 3 -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ V
42, 3ifex 4579 . 2 if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) ∈ V
51, 4eqeltri 2830 1 ⨜[𝐴𝐵]𝐶 d𝑥 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3475  ifcif 4529   class class class wbr 5149  (class class class)co 7409  cle 11249  -cneg 11445  (,)cioo 13324  citg 25135  cdit 25363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-uni 4910  df-iota 6496  df-fv 6552  df-ov 7412  df-neg 11447  df-sum 15633  df-itg 25140  df-ditg 25364
This theorem is referenced by:  itgsubstlem  25565
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