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Theorem dtruarb 4175
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 4541 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 4162 . . 3  |-  E. x  z  e.  x
2 ax-nul 4113 . . . 4  |-  E. y A. z  -.  z  e.  y
3 sp 1504 . . . 4  |-  ( A. z  -.  z  e.  y  ->  -.  z  e.  y )
42, 3eximii 1595 . . 3  |-  E. y  -.  z  e.  y
5 eeanv 1925 . . 3  |-  ( E. x E. y ( z  e.  x  /\  -.  z  e.  y
)  <->  ( E. x  z  e.  x  /\  E. y  -.  z  e.  y ) )
61, 4, 5mpbir2an 937 . 2  |-  E. x E. y ( z  e.  x  /\  -.  z  e.  y )
7 nelneq2 2272 . . 3  |-  ( ( z  e.  x  /\  -.  z  e.  y
)  ->  -.  x  =  y )
872eximi 1594 . 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 1346   E.wex 1485
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-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4113  ax-pow 4158
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166
This theorem is referenced by: (None)
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