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

Theorem exsnrex 4000
 Description: There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)
Assertion
Ref Expression
exsnrex

Proof of Theorem exsnrex
StepHypRef Expression
1 ssnid 3989 . . . . 5
2 eleq2 2538 . . . . 5
31, 2mpbiri 241 . . . 4
43pm4.71ri 645 . . 3
54exbii 1726 . 2
6 df-rex 2762 . 2
75, 6bitr4i 260 1
 Colors of variables: wff setvar class Syntax hints:   wb 189   wa 376   wceq 1452  wex 1671   wcel 1904  wrex 2757  csn 3959 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-v 3033  df-sn 3960 This theorem is referenced by:  frgrawopreg1  25857  frgrawopreg2  25858
 Copyright terms: Public domain W3C validator