ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  uni0c GIF version

Theorem uni0c 3685
Description: The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
uni0c ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem uni0c
StepHypRef Expression
1 uni0b 3684 . 2 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
2 dfss3 3016 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
3 velsn 3467 . . 3 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43ralbii 2385 . 2 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
51, 2, 43bitri 205 1 ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1290  wcel 1439  wral 2360  wss 3000  c0 3287  {csn 3450   cuni 3659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-in 3006  df-ss 3013  df-nul 3288  df-sn 3456  df-uni 3660
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator