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Theorem dtruarb 3969
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 4310 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 3958 . . 3 𝑥 𝑧𝑥
2 ax-nul 3910 . . . 4 𝑦𝑧 ¬ 𝑧𝑦
3 sp 1417 . . . 4 (∀𝑧 ¬ 𝑧𝑦 → ¬ 𝑧𝑦)
42, 3eximii 1509 . . 3 𝑦 ¬ 𝑧𝑦
5 eeanv 1823 . . 3 (∃𝑥𝑦(𝑧𝑥 ∧ ¬ 𝑧𝑦) ↔ (∃𝑥 𝑧𝑥 ∧ ∃𝑦 ¬ 𝑧𝑦))
61, 4, 5mpbir2an 860 . 2 𝑥𝑦(𝑧𝑥 ∧ ¬ 𝑧𝑦)
7 nelneq2 2155 . . 3 ((𝑧𝑥 ∧ ¬ 𝑧𝑦) → ¬ 𝑥 = 𝑦)
872eximi 1508 . 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 101  wal 1257  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038  ax-nul 3910  ax-pow 3954
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052
This theorem is referenced by: (None)
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