Theorem dtru 4513

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 4512. (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 4512	. 2 |- E. x -. x = y
2		exnalim 1623	. 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 1330   E.wex 1469

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-setind 4490

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-v 2711  df-dif 3100  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562

This theorem is referenced by:  oprabidlem  5842