![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ruALT | GIF version |
Description: Alternate proof of Russell's Paradox ru 2976, simplified using (indirectly) the Axiom of Set Induction ax-setind 4554. (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 4150 | . . 3 ⊢ ¬ V ∈ V | |
2 | df-nel 2456 | . . 3 ⊢ (V ∉ V ↔ ¬ V ∈ V) | |
3 | 1, 2 | mpbir 146 | . 2 ⊢ V ∉ V |
4 | ruv 4567 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | |
5 | neleq1 2459 | . . 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 146 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 105 = wceq 1364 ∈ wcel 2160 {cab 2175 ∉ wnel 2455 Vcvv 2752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-v 2754 df-dif 3146 df-sn 3613 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |