Theorem dtru 4374

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 4373. (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 4373	. 2 |- E. x -. x = y
2		exnalim 1582	. 2 |- ( E. x -. x = y -> -. A. x x = y )
3	1, 2	ax-mp 7	1 |- -. A. x x = y

Syntax hints:   -. wn 3   A.wal 1287   E.wex 1426

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-setind 4351

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450

This theorem is referenced by:  oprabidlem  5672