Theorem dtrucor3 49111

Description: An example of how ax-5 1912 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5387 in the ZF set theory. axc16nf 2271 and euae 2661 demonstrate that the violation of dtru 5387 leads to a model with only one object assuming its existence (ax-6 1969). The conclusion is also provable in the empty model ( see emptyal 1910). See also nf5 2289 and nf5i 2152 for the relation between unconditional ax-5 1912 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

Step	Hyp	Ref	Expression
1		ax6ev 1971	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		dtrucor3.1	. . . 4 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
3		dtrucor3.2	. . . 4 ⊢ (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
4	2, 3	mto 197	. . 3 ⊢ ¬ 𝑥 = 𝑦
5	4	nex 1802	. 2 ⊢ ¬ ∃𝑥 𝑥 = 𝑦
6	1, 5	pm2.24ii 120	1 ⊢ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1540  ∃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 1969

This theorem depends on definitions:  df-bi 207  df-ex 1782

This theorem is referenced by: (None)