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Theorem dtruarb 3970
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4311 in which we are given a set  y and go from there to a set  x which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)
Assertion
Ref Expression
dtruarb  |-  E. x E. y  -.  x  =  y
Distinct variable group:    x, y

Proof of Theorem dtruarb
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 el 3959 . . 3  |-  E. x  z  e.  x
2 ax-nul 3911 . . . 4  |-  E. y A. z  -.  z  e.  y
3 sp 1417 . . . 4  |-  ( A. z  -.  z  e.  y  ->  -.  z  e.  y )
42, 3eximii 1509 . . 3  |-  E. y  -.  z  e.  y
5 eeanv 1823 . . 3  |-  ( E. x E. y ( z  e.  x  /\  -.  z  e.  y
)  <->  ( E. x  z  e.  x  /\  E. y  -.  z  e.  y ) )
61, 4, 5mpbir2an 860 . 2  |-  E. x E. y ( z  e.  x  /\  -.  z  e.  y )
7 nelneq2 2155 . . 3  |-  ( ( z  e.  x  /\  -.  z  e.  y
)  ->  -.  x  =  y )
872eximi 1508 . 2  |-  ( E. x E. y ( z  e.  x  /\  -.  z  e.  y
)  ->  E. x E. y  -.  x  =  y )
96, 8ax-mp 7 1  |-  E. x E. y  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 101   A.wal 1257   E.wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038  ax-nul 3911  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator