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Theorem dtrucor3 47572
Description: An example of how ax-5 1913 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5436 in the ZF set theory. axc16nf 2254 and euae 2655 demonstrate that the violation of dtru 5436 leads to a model with only one object assuming its existence (ax-6 1971). The conclusion is also provable in the empty model ( see emptyal 1911). See also nf5 2278 and nf5i 2142 for the relation between unconditional ax-5 1913 and being not free. (Contributed by Zhi Wang, 23-Sep-2024.)
Hypotheses
Ref Expression
dtrucor3.1 ¬ ∀𝑥 𝑥 = 𝑦
dtrucor3.2 (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
Assertion
Ref Expression
dtrucor3 𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem dtrucor3
StepHypRef Expression
1 ax6ev 1973 . 2 𝑥 𝑥 = 𝑦
2 dtrucor3.1 . . . 4 ¬ ∀𝑥 𝑥 = 𝑦
3 dtrucor3.2 . . . 4 (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
42, 3mto 196 . . 3 ¬ 𝑥 = 𝑦
54nex 1802 . 2 ¬ ∃𝑥 𝑥 = 𝑦
61, 5pm2.24ii 120 1 𝑥 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1539  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-6 1971
This theorem depends on definitions:  df-bi 206  df-ex 1782
This theorem is referenced by: (None)
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