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Mirrors > Home > MPE Home > Th. List > pw0 | Structured version Visualization version 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 3973 | . . 3 ⊢ (𝑥 ⊆ ∅ ↔ 𝑥 = ∅) | |
2 | 1 | abbii 2739 | . 2 ⊢ {𝑥 ∣ 𝑥 ⊆ ∅} = {𝑥 ∣ 𝑥 = ∅} |
3 | df-pw 4160 | . 2 ⊢ 𝒫 ∅ = {𝑥 ∣ 𝑥 ⊆ ∅} | |
4 | df-sn 4178 | . 2 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
5 | 2, 3, 4 | 3eqtr4i 2654 | 1 ⊢ 𝒫 ∅ = {∅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 {cab 2608 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 |
This theorem is referenced by: p0ex 4853 pwfi 8261 ackbij1lem14 9055 fin1a2lem12 9233 0tsk 9577 hashbc 13237 incexclem 14568 sn0topon 20802 sn0cld 20894 ust0 22023 uhgr0vb 25967 uhgr0 25968 esumnul 30110 rankeq1o 32278 ssoninhaus 32447 sge00 40593 |
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