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Mirrors > Home > ILE Home > Th. List > setind2 | GIF version |
Description: Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.) |
Ref | Expression |
---|---|
setind2 | ⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwss 3397 | . 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) | |
2 | setind 4282 | . 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V) | |
3 | 1, 2 | sylbi 119 | 1 ⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 𝒫 cpw 3382 |
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-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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-ral 2353 df-v 2603 df-in 2979 df-ss 2986 df-pw 3384 |
This theorem is referenced by: (None) |
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