Theorem ruvALT 41459

Description: Alternate proof of ruv 9597 with one fewer syntax step thanks to using elirrv 9591 instead of elirr 9592. 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 29684. (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 3479	. . . 4 ⊢ 𝑥 ∈ V
2		elirrv 9591	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3050	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 264	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2870	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2742	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1542   ∈ wcel 2107  {cab 2710   ∉ wnel 3047  Vcvv 3475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-pr 5428  ax-reg 9587

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nel 3048  df-ral 3063  df-rex 3072  df-v 3477  df-un 3954  df-sn 4630  df-pr 4632

This theorem is referenced by: (None)