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

Theorem uni0c 3596
 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 = ∅ ↔ x A x = ∅)
Distinct variable group:   x,A

Proof of Theorem uni0c
StepHypRef Expression
1 uni0b 3595 . 2 ( A = ∅ ↔ A ⊆ {∅})
2 dfss3 2929 . 2 (A ⊆ {∅} ↔ x A x {∅})
3 elsn 3381 . . 3 (x {∅} ↔ x = ∅)
43ralbii 2324 . 2 (x A x {∅} ↔ x A x = ∅)
51, 2, 43bitri 195 1 ( A = ∅ ↔ x A x = ∅)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911  ∅c0 3218  {csn 3366  ∪ cuni 3570 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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3372  df-uni 3571 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator