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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 3341 | . . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅) | |
2 | 1 | abbii 2210 | . 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅} |
3 | df-pw 3451 | . 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅} | |
4 | df-sn 3472 | . 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
5 | 2, 3, 4 | 3eqtr4i 2125 | 1 ⊢ 𝒫 ∅ = {∅} |
Colors of variables: wff set class |
Syntax hints: = wceq 1296 {cab 2081 ⊆ wss 3013 ∅c0 3302 𝒫 cpw 3449 {csn 3466 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-dif 3015 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 |
This theorem is referenced by: p0ex 4044 sn0topon 11940 sn0cld 11989 |
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