Theorem dtru 4608

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 4607. (Contributed by Jim Kingdon, 29-Dec-2018.)

Assertion

Ref	Expression
dtru	⊢ ¬ ∀𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem dtru

Step	Hyp	Ref	Expression
1		dtruex 4607	. 2 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
2		exnalim 1669	. 2 ⊢ (∃𝑥 ¬ 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3	1, 2	ax-mp 5	1 ⊢ ¬ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1371  ∃wex 1515

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639

This theorem is referenced by:  oprabidlem  5975