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Theorem dtruarb 4017
Description: At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4365 in which we are given a set 𝑦 and go from there to a set 𝑥 which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)
Assertion
Ref Expression
dtruarb 𝑥𝑦 ¬ 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruarb
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 el 4005 . . 3 𝑥 𝑧𝑥
2 ax-nul 3957 . . . 4 𝑦𝑧 ¬ 𝑧𝑦
3 sp 1446 . . . 4 (∀𝑧 ¬ 𝑧𝑦 → ¬ 𝑧𝑦)
42, 3eximii 1538 . . 3 𝑦 ¬ 𝑧𝑦
5 eeanv 1855 . . 3 (∃𝑥𝑦(𝑧𝑥 ∧ ¬ 𝑧𝑦) ↔ (∃𝑥 𝑧𝑥 ∧ ∃𝑦 ¬ 𝑧𝑦))
61, 4, 5mpbir2an 888 . 2 𝑥𝑦(𝑧𝑥 ∧ ¬ 𝑧𝑦)
7 nelneq2 2189 . . 3 ((𝑧𝑥 ∧ ¬ 𝑧𝑦) → ¬ 𝑥 = 𝑦)
872eximi 1537 . 2 (∃𝑥𝑦(𝑧𝑥 ∧ ¬ 𝑧𝑦) → ∃𝑥𝑦 ¬ 𝑥 = 𝑦)
96, 8ax-mp 7 1 𝑥𝑦 ¬ 𝑥 = 𝑦
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wal 1287  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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070  ax-nul 3957  ax-pow 4001
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-cleq 2081  df-clel 2084
This theorem is referenced by: (None)
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