![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ruALT | GIF version |
Description: Alternate proof of Russell's Paradox ru 2912, simplified using (indirectly) the Axiom of Set Induction ax-setind 4460. (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 4068 | . . 3 ⊢ ¬ V ∈ V | |
2 | df-nel 2405 | . . 3 ⊢ (V ∉ V ↔ ¬ V ∈ V) | |
3 | 1, 2 | mpbir 145 | . 2 ⊢ V ∉ V |
4 | ruv 4473 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | |
5 | neleq1 2408 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} = V → ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V)) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V) |
7 | 3, 6 | mpbir 145 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 = wceq 1332 ∈ wcel 1481 {cab 2126 ∉ wnel 2404 Vcvv 2689 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-v 2691 df-dif 3078 df-sn 3538 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |