Theorem ruvALT 42672

Description: Alternate proof of ruv 9649 with one fewer syntax step thanks to using elirrv 9643 instead of elirr 9644. 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 30445. (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 3485	. . . 4 ⊢ 𝑥 ∈ V
2		elirrv 9643	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3049	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 264	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2877	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2746	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1539   ∈ wcel 2108  {cab 2714   ∉ wnel 3046  Vcvv 3481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-pr 5441  ax-reg 9639

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nel 3047  df-ral 3062  df-rex 3071  df-v 3483  df-un 3971  df-sn 4635  df-pr 4637

This theorem is referenced by: (None)