MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixp0 Unicode version

Theorem ixp0 6849
Description: The infinite Cartesian product of a family  B ( x ) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 8110. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
ixp0  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )

Proof of Theorem ixp0
StepHypRef Expression
1 nne 2450 . . . 4  |-  ( -.  B  =/=  (/)  <->  B  =  (/) )
21rexbii 2568 . . 3  |-  ( E. x  e.  A  -.  B  =/=  (/)  <->  E. x  e.  A  B  =  (/) )
3 rexnal 2554 . . 3  |-  ( E. x  e.  A  -.  B  =/=  (/)  <->  -.  A. x  e.  A  B  =/=  (/) )
42, 3bitr3i 242 . 2  |-  ( E. x  e.  A  B  =  (/)  <->  -.  A. x  e.  A  B  =/=  (/) )
5 ixpn0 6848 . . 3  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
65necon1bi 2489 . 2  |-  ( -. 
A. x  e.  A  B  =/=  (/)  ->  X_ x  e.  A  B  =  (/) )
74, 6sylbi 187 1  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455   X_cixp 6817
This theorem is referenced by:  ac9  8110  ac9s  8120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-nul 3456  df-ixp 6818
  Copyright terms: Public domain W3C validator