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Mirrors > Home > ILE Home > Th. List > ruALT | GIF version |
Description: Alternate proof of Russell's Paradox ru 2814, simplified using (indirectly) the Axiom of Set Induction ax-setind 4280. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ruALT | ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vprc 3909 | . . 3 ⊢ ¬ V ∈ V | |
2 | df-nel 2340 | . . 3 ⊢ (V ∉ V ↔ ¬ V ∈ V) | |
3 | 1, 2 | mpbir 144 | . 2 ⊢ V ∉ V |
4 | ruv 4293 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | |
5 | neleq1 2343 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} = V → ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V)) | |
6 | 4, 5 | ax-mp 7 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V) |
7 | 3, 6 | mpbir 144 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 = wceq 1284 ∈ wcel 1433 {cab 2067 ∉ wnel 2339 Vcvv 2601 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-v 2603 df-dif 2975 df-sn 3404 |
This theorem is referenced by: (None) |
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