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Mirrors > Home > ILE Home > Th. List > ruv | 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 | df-v 2683 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
2 | equid 1677 | . . . 4 ⊢ 𝑥 = 𝑥 | |
3 | elirrv 4458 | . . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥 | |
4 | 3 | nelir 2404 | . . . 4 ⊢ 𝑥 ∉ 𝑥 |
5 | 2, 4 | 2th 173 | . . 3 ⊢ (𝑥 = 𝑥 ↔ 𝑥 ∉ 𝑥) |
6 | 5 | abbii 2253 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝑥 ∉ 𝑥} |
7 | 1, 6 | eqtr2i 2159 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 {cab 2123 ∉ wnel 2401 Vcvv 2681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-v 2683 df-dif 3068 df-sn 3528 |
This theorem is referenced by: ruALT 4461 |
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