Theorem ruvALT 42657

Description: Alternate proof of ruv 9555 with one fewer syntax step thanks to using elirrv 9549 instead of elirr 9550. 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 30329. (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 3451	. . . 4 ⊢ 𝑥 ∈ V
2		elirrv 9549	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3032	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 264	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2863	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2738	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1540   ∈ wcel 2109  {cab 2707   ∉ wnel 3029  Vcvv 3447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-pr 5387  ax-reg 9545

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nel 3030  df-ral 3045  df-rex 3054  df-v 3449  df-un 3919  df-sn 4590  df-pr 4592

This theorem is referenced by: (None)