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Theorem dtruex 4543
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4177 can also be summarized as "at least two sets exist", the difference is that dtruarb 4177 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific  y, we can construct a set  x which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
dtruex  |-  E. x  -.  x  =  y
Distinct variable group:    x, y

Proof of Theorem dtruex
StepHypRef Expression
1 vex 2733 . . . . 5  |-  y  e. 
_V
21snex 4171 . . . 4  |-  { y }  e.  _V
32isseti 2738 . . 3  |-  E. x  x  =  { y }
4 elirrv 4532 . . . . . . 7  |-  -.  y  e.  y
5 vsnid 3615 . . . . . . . 8  |-  y  e. 
{ y }
6 eleq2 2234 . . . . . . . 8  |-  ( y  =  { y }  ->  ( y  e.  y  <->  y  e.  {
y } ) )
75, 6mpbiri 167 . . . . . . 7  |-  ( y  =  { y }  ->  y  e.  y )
84, 7mto 657 . . . . . 6  |-  -.  y  =  { y }
9 eqtr 2188 . . . . . 6  |-  ( ( y  =  x  /\  x  =  { y } )  ->  y  =  { y } )
108, 9mto 657 . . . . 5  |-  -.  (
y  =  x  /\  x  =  { y } )
11 ancom 264 . . . . 5  |-  ( ( y  =  x  /\  x  =  { y } )  <->  ( x  =  { y }  /\  y  =  x )
)
1210, 11mtbi 665 . . . 4  |-  -.  (
x  =  { y }  /\  y  =  x )
1312imnani 686 . . 3  |-  ( x  =  { y }  ->  -.  y  =  x )
143, 13eximii 1595 . 2  |-  E. x  -.  y  =  x
15 equcom 1699 . . . 4  |-  ( y  =  x  <->  x  =  y )
1615notbii 663 . . 3  |-  ( -.  y  =  x  <->  -.  x  =  y )
1716exbii 1598 . 2  |-  ( E. x  -.  y  =  x  <->  E. x  -.  x  =  y )
1814, 17mpbi 144 1  |-  E. x  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    = wceq 1348   E.wex 1485    e. wcel 2141   {csn 3583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589
This theorem is referenced by:  dtru  4544  eunex  4545  brprcneu  5489
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