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Theorem dtruex 4442
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 4083 can also be summarized as "at least two sets exist", the difference is that dtruarb 4083 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 2661 . . . . 5  |-  y  e. 
_V
21snex 4077 . . . 4  |-  { y }  e.  _V
32isseti 2666 . . 3  |-  E. x  x  =  { y }
4 elirrv 4431 . . . . . . 7  |-  -.  y  e.  y
5 vsnid 3525 . . . . . . . 8  |-  y  e. 
{ y }
6 eleq2 2179 . . . . . . . 8  |-  ( y  =  { y }  ->  ( y  e.  y  <->  y  e.  {
y } ) )
75, 6mpbiri 167 . . . . . . 7  |-  ( y  =  { y }  ->  y  e.  y )
84, 7mto 634 . . . . . 6  |-  -.  y  =  { y }
9 eqtr 2133 . . . . . 6  |-  ( ( y  =  x  /\  x  =  { y } )  ->  y  =  { y } )
108, 9mto 634 . . . . 5  |-  -.  (
y  =  x  /\  x  =  { y } )
11 ancom 264 . . . . 5  |-  ( ( y  =  x  /\  x  =  { y } )  <->  ( x  =  { y }  /\  y  =  x )
)
1210, 11mtbi 642 . . . 4  |-  -.  (
x  =  { y }  /\  y  =  x )
1312imnani 663 . . 3  |-  ( x  =  { y }  ->  -.  y  =  x )
143, 13eximii 1564 . 2  |-  E. x  -.  y  =  x
15 equcom 1665 . . . 4  |-  ( y  =  x  <->  x  =  y )
1615notbii 640 . . 3  |-  ( -.  y  =  x  <->  -.  x  =  y )
1716exbii 1567 . 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 1314   E.wex 1451    e. wcel 1463   {csn 3495
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501
This theorem is referenced by:  dtru  4443  eunex  4444  brprcneu  5380
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