Theorem ruvALT 43461

Description: Alternate proof of ruv 9577 with one fewer syntax step thanks to using elirrv 9566 instead of elirr 9569. 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 30824. (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 3461	. . . 4 ⊢ 𝑥 ∈ V
2		elirrv 9566	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3069	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 267	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2900	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2774	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  {cab 2743   ∉ wnel 3066  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-reg 9561

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nel 3067  df-v 3459

This theorem is used by: (None)