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Theorem dtruex 4237
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Although dtruarb 3933 can also be summarized as "at least two sets exist", the difference is that dtruarb 3933 shows the existence of two sets which are not equal to each other, but this theorem says that given a specific , we can construct a set which does not equal it. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
dtruex
Distinct variable group:   ,

Proof of Theorem dtruex
StepHypRef Expression
1 vex 2554 . . . . 5 
_V
2 snexgOLD 3926 . . . . 5  _V  { }  _V
31, 2ax-mp 7 . . . 4  { }  _V
4 isset 2555 . . . 4  { }  _V  { }
53, 4mpbi 133 . . 3  { }
6 elirr 4224 . . . . . . . 8
7 ssnid 3395 . . . . . . . . 9 
{ }
8 eleq2 2098 . . . . . . . . 9  { }  { }
97, 8mpbiri 157 . . . . . . . 8  { }
106, 9mto 587 . . . . . . 7  { }
11 eqtr 2054 . . . . . . 7  { }  { }
1210, 11mto 587 . . . . . 6  { }
13 ancom 253 . . . . . 6  { }  { }
1412, 13mtbi 594 . . . . 5  { }
1514imnani 624 . . . 4  { }
1615eximi 1488 . . 3  { }
175, 16ax-mp 7 . 2
18 eqcom 2039 . . . 4
1918notbii 593 . . 3
2019exbii 1493 . 2
2117, 20mpbi 133 1
Colors of variables: wff set class
Syntax hints:   wn 3   wa 97   wceq 1242  wex 1378   wcel 1390   _Vcvv 2551   {csn 3367
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373
This theorem is referenced by:  dtru  4238  eunex  4239  brprcneu  5114
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