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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 2621 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
2 | equid 1634 | . . . 4 ⊢ 𝑥 = 𝑥 | |
3 | elirrv 4362 | . . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥 | |
4 | 3 | nelir 2353 | . . . 4 ⊢ 𝑥 ∉ 𝑥 |
5 | 2, 4 | 2th 172 | . . 3 ⊢ (𝑥 = 𝑥 ↔ 𝑥 ∉ 𝑥) |
6 | 5 | abbii 2203 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝑥 ∉ 𝑥} |
7 | 1, 6 | eqtr2i 2109 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 {cab 2074 ∉ wnel 2350 Vcvv 2619 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-setind 4351 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-v 2621 df-dif 3001 df-sn 3450 |
This theorem is referenced by: ruALT 4365 |
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