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Theorem dtruarb 4085
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 4444 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 4072 . . 3  |-  E. x  z  e.  x
2 ax-nul 4024 . . . 4  |-  E. y A. z  -.  z  e.  y
3 sp 1473 . . . 4  |-  ( A. z  -.  z  e.  y  ->  -.  z  e.  y )
42, 3eximii 1566 . . 3  |-  E. y  -.  z  e.  y
5 eeanv 1884 . . 3  |-  ( E. x E. y ( z  e.  x  /\  -.  z  e.  y
)  <->  ( E. x  z  e.  x  /\  E. y  -.  z  e.  y ) )
61, 4, 5mpbir2an 911 . 2  |-  E. x E. y ( z  e.  x  /\  -.  z  e.  y )
7 nelneq2 2219 . . 3  |-  ( ( z  e.  x  /\  -.  z  e.  y
)  ->  -.  x  =  y )
872eximi 1565 . 2  |-  ( E. x E. y ( z  e.  x  /\  -.  z  e.  y
)  ->  E. x E. y  -.  x  =  y )
96, 8ax-mp 5 1  |-  E. x E. y  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103   A.wal 1314   E.wex 1453
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 588  ax-in2 589  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099  ax-nul 4024  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-cleq 2110  df-clel 2113
This theorem is referenced by: (None)
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