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Theorem dtru 4305
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). If we assumed the law of the excluded middle this would be equivalent to dtruex 4304. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
dtru  |-  -.  A. x  x  =  y
Distinct variable group:    x, y

Proof of Theorem dtru
StepHypRef Expression
1 dtruex 4304 . 2  |-  E. x  -.  x  =  y
2 exnalim 1578 . 2  |-  ( E. x  -.  x  =  y  ->  -.  A. x  x  =  y )
31, 2ax-mp 7 1  |-  -.  A. x  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1283   E.wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-setind 4282
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406
This theorem is referenced by:  oprabidlem  5561
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