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Theorem dtru 4577
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 4576. (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 4576 . 2  |-  E. x  -.  x  =  y
2 exnalim 1657 . 2  |-  ( E. x  -.  x  =  y  ->  -.  A. x  x  =  y )
31, 2ax-mp 5 1  |-  -.  A. x  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1362   E.wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613
This theorem is referenced by:  oprabidlem  5927
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