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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 2732 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
2 | equid 1694 | . . . 4 ⊢ 𝑥 = 𝑥 | |
3 | elirrv 4532 | . . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥 | |
4 | 3 | nelir 2438 | . . . 4 ⊢ 𝑥 ∉ 𝑥 |
5 | 2, 4 | 2th 173 | . . 3 ⊢ (𝑥 = 𝑥 ↔ 𝑥 ∉ 𝑥) |
6 | 5 | abbii 2286 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝑥 ∉ 𝑥} |
7 | 1, 6 | eqtr2i 2192 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 {cab 2156 ∉ wnel 2435 Vcvv 2730 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-v 2732 df-dif 3123 df-sn 3589 |
This theorem is referenced by: ruALT 4535 |
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