Theorem ruvALT 42772

Description: Alternate proof of ruv 9491 with one fewer syntax step thanks to using elirrv 9483 instead of elirr 9485. 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 30380. (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 3440	. . . 4 ⊢ 𝑥 ∈ V
2		elirrv 9483	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3035	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 264	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2866	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2740	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1541   ∈ wcel 2111  {cab 2709   ∉ wnel 3032  Vcvv 3436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-pr 5368  ax-reg 9478

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nel 3033  df-v 3438

This theorem is referenced by: (None)