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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 2659 | . 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} | |
2 | equid 1660 | . . . 4 ⊢ 𝑥 = 𝑥 | |
3 | elirrv 4423 | . . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥 | |
4 | 3 | nelir 2380 | . . . 4 ⊢ 𝑥 ∉ 𝑥 |
5 | 2, 4 | 2th 173 | . . 3 ⊢ (𝑥 = 𝑥 ↔ 𝑥 ∉ 𝑥) |
6 | 5 | abbii 2230 | . 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝑥 ∉ 𝑥} |
7 | 1, 6 | eqtr2i 2136 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1314 {cab 2101 ∉ wnel 2377 Vcvv 2657 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-setind 4412 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-v 2659 df-dif 3039 df-sn 3499 |
This theorem is referenced by: ruALT 4426 |
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