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Mirrors > Home > MPE Home > Th. List > Mathboxes > ruvALT | Structured version Visualization version GIF version |
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 29653. (Contributed by SN, 1-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
ruvALT | ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V |
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 |
Colors of variables: wff setvar class |
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) |
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