Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  iun0 Structured version   GIF version

Theorem iun0 3683
 Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iun0 x A ∅ = ∅

Proof of Theorem iun0
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 noel 3201 . . . . . 6 ¬ y
21a1i 9 . . . . 5 (x A → ¬ y ∅)
32nrex 2385 . . . 4 ¬ x A y
4 eliun 3631 . . . 4 (y x A ∅ ↔ x A y ∅)
53, 4mtbir 583 . . 3 ¬ y x A
65, 12false 604 . 2 (y x A ∅ ↔ y ∅)
76eqriv 2015 1 x A ∅ = ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1226   ∈ wcel 1370  ∃wrex 2281  ∅c0 3197  ∪ ciun 3627 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-nul 3198  df-iun 3629 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator