Theorem ruvALT 40096

Description: Alternate proof of ruv 9291 with one fewer syntax step thanks to using elirrv 9285 instead of elirr 9286. 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 28665. (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 3426	. . . 4 ⊢ 𝑥 ∈ V
2		elirrv 9285	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3051	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 263	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	abbi2i 2878	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2747	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1539   ∈ wcel 2108  {cab 2715   ∉ wnel 3048  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-reg 9281

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nel 3049  df-ral 3068  df-rex 3069  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-sn 4559  df-pr 4561

This theorem is referenced by: (None)