Theorem dtrucor3 50114

Description: An example of how ax-5 1945 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5409 in the ZF set theory. axc16nf 2301 and euae 2687 demonstrate that the violation of dtru 5409 leads to a model with only one object assuming its existence (ax-6 2002). The conclusion is also provable in the empty model ( see emptyal 1943). See also nf5 2319 and nf5i 2186 for the relation between unconditional ax-5 1945 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 2004	. 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 1833	. 2 ⊢ ¬ ∃𝑥 𝑥 = 𝑦
6	1, 5	pm2.24ii 121	1 ⊢ ∀𝑥 𝑥 = 𝑦

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-6 2002

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by: (None)