Theorem dtrucor3 49985

Description: An example of how ax-5 1942 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5404 in the ZF set theory. axc16nf 2297 and euae 2684 demonstrate that the violation of dtru 5404 leads to a model with only one object assuming its existence (ax-6 1999). The conclusion is also provable in the empty model ( see emptyal 1940). See also nf5 2315 and nf5i 2182 for the relation between unconditional ax-5 1942 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 2001	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
2		dtrucor3.1	. . . 4 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
3		dtrucor3.2	. . . 4 ⊢ (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)
4	2, 3	mto 200	. . 3 ⊢ ¬ 𝑥 = 𝑦
5	4	nex 1832	. 2 ⊢ ¬ ∃𝑥 𝑥 = 𝑦
6	1, 5	pm2.24ii 121	1 ⊢ ∀𝑥 𝑥 = 𝑦

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-6 1999

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by: (None)