Theorem dtrucor3 48648

Description: An example of how ax-5 1908 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5447 in the ZF set theory. axc16nf 2261 and euae 2658 demonstrate that the violation of dtru 5447 leads to a model with only one object assuming its existence (ax-6 1965). The conclusion is also provable in the empty model ( see emptyal 1906). See also nf5 2281 and nf5i 2144 for the relation between unconditional ax-5 1908 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 1967	. 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 1797	. 2 ⊢ ¬ ∃𝑥 𝑥 = 𝑦
6	1, 5	pm2.24ii 120	1 ⊢ ∀𝑥 𝑥 = 𝑦

Syntax hints:  ¬ wn 3  ∀wal 1535  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-6 1965

This theorem depends on definitions:  df-bi 207  df-ex 1777

This theorem is referenced by: (None)