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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 3496 | . 2 ⊢ (𝒫 𝐴 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) | |
2 | setind 4424 | . 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V) | |
3 | 1, 2 | sylbi 120 | 1 ⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1314 = wceq 1316 ∈ wcel 1465 Vcvv 2660 ⊆ wss 3041 𝒫 cpw 3480 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-ral 2398 df-v 2662 df-in 3047 df-ss 3054 df-pw 3482 |
This theorem is referenced by: (None) |
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