Theorem ruvALT 40620

Description: Alternate proof of ruv 9538 with one fewer syntax step thanks to using elirrv 9532 instead of elirr 9533. 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 29344. (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 3449	. . . 4 ⊢ 𝑥 ∈ V
2		elirrv 9532	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3052	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 263	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	abbi2i 2873	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2745	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1541   ∈ wcel 2106  {cab 2713   ∉ wnel 3049  Vcvv 3445

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 2707  ax-sep 5256  ax-pr 5384  ax-reg 9528

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 2714  df-cleq 2728  df-clel 2814  df-nel 3050  df-ral 3065  df-rex 3074  df-v 3447  df-un 3915  df-sn 4587  df-pr 4589

This theorem is referenced by: (None)