Theorem ruvALT 43518

Description: Alternate proof of ruv 9583 with one fewer syntax step thanks to using elirrv 9572 instead of elirr 9575. 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 30883. (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 3454	. . . 4 ⊢ 𝑥 ∈ V
2		elirrv 9572	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3064	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 267	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2895	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2769	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  {cab 2738   ∉ wnel 3061  Vcvv 3450

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-reg 9567

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nel 3062  df-v 3452

This theorem is used by: (None)