Theorem ruvALT 43429

Description: Alternate proof of ruv 9568 with one fewer syntax step thanks to using elirrv 9557 instead of elirr 9560. However, it does not change the compressed proof size or the number of symbols in the generated display, so it is not considered a shortening according to conventions 30762. (Contributed by SN, 1-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ruvALT	⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Proof of Theorem ruvALT

Step	Hyp	Ref	Expression
1		vex 3458	. . . 4 ⊢ 𝑥 ∈ V
2		elirrv 9557	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3066	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 267	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2897	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2771	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2142  {cab 2740   ∉ wnel 3063  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-reg 9552

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nel 3064  df-v 3456

This theorem is used by: (None)