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Mirrors > Home > MPE Home > Th. List > ruv | Structured version Visualization version GIF version |
Description: The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.) |
Ref | Expression |
---|---|
ruv | ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3449 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | elirr 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 |
Colors of variables: wff setvar class |
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 9527 |
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: ruALT 9538 |
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