Theorem dtru 4629

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 4628. (Contributed by Jim Kingdon, 29-Dec-2018.)

Assertion

Ref	Expression
dtru	|- -. A. x x = y

Distinct variable group:   x, y

Proof of Theorem dtru

Step	Hyp	Ref	Expression
1		dtruex 4628	. 2 |- E. x -. x = y
2		exnalim 1672	. 2 |- ( E. x -. x = y -> -. A. x x = y )
3	1, 2	ax-mp 5	1 |- -. A. x x = y

Syntax hints:   -. wn 3   A.wal 1373   E.wex 1518

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-v 2781  df-dif 3179  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652

This theorem is referenced by:  oprabidlem  6005