Theorem ruvALT 40693

Description: Alternate proof of ruv 9547 with one fewer syntax step thanks to using elirrv 9541 instead of elirr 9542. 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 29407. (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 3450	. . . 4 ⊢ 𝑥 ∈ V
2		elirrv 9541	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3048	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 263	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2868	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2740	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1541   ∈ wcel 2106  {cab 2708   ∉ wnel 3045  Vcvv 3446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-pr 5389  ax-reg 9537

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nel 3046  df-ral 3061  df-rex 3070  df-v 3448  df-un 3918  df-sn 4592  df-pr 4594

This theorem is referenced by: (None)