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Theorem dtruarb 4119
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 4478 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 4106 . . 3 𝑥 𝑧𝑥
2 ax-nul 4058 . . . 4 𝑦𝑧 ¬ 𝑧𝑦
3 sp 1489 . . . 4 (∀𝑧 ¬ 𝑧𝑦 → ¬ 𝑧𝑦)
42, 3eximii 1582 . . 3 𝑦 ¬ 𝑧𝑦
5 eeanv 1905 . . 3 (∃𝑥𝑦(𝑧𝑥 ∧ ¬ 𝑧𝑦) ↔ (∃𝑥 𝑧𝑥 ∧ ∃𝑦 ¬ 𝑧𝑦))
61, 4, 5mpbir2an 927 . 2 𝑥𝑦(𝑧𝑥 ∧ ¬ 𝑧𝑦)
7 nelneq2 2242 . . 3 ((𝑧𝑥 ∧ ¬ 𝑧𝑦) → ¬ 𝑥 = 𝑦)
872eximi 1581 . 2 (∃𝑥𝑦(𝑧𝑥 ∧ ¬ 𝑧𝑦) → ∃𝑥𝑦 ¬ 𝑥 = 𝑦)
96, 8ax-mp 5 1 𝑥𝑦 ¬ 𝑥 = 𝑦
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wal 1330  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-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122  ax-nul 4058  ax-pow 4102
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-cleq 2133  df-clel 2136
This theorem is referenced by: (None)
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