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Mirrors > Home > ILE Home > Th. List > pw0 | GIF version |
Description: Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
pw0 | ⊢ 𝒫 ∅ = {∅} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss0b 3307 | . . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅) | |
2 | 1 | abbii 2200 | . 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅} |
3 | df-pw 3411 | . 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅} | |
4 | df-sn 3431 | . 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
5 | 2, 3, 4 | 3eqtr4i 2115 | 1 ⊢ 𝒫 ∅ = {∅} |
Colors of variables: wff set class |
Syntax hints: = wceq 1287 {cab 2071 ⊆ wss 2986 ∅c0 3272 𝒫 cpw 3409 {csn 3425 |
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 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-v 2616 df-dif 2988 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 |
This theorem is referenced by: p0ex 3990 |
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