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Mirrors > Home > NFE Home > Th. List > pw0 | GIF version |
Description: Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (The proof was shortened by Andrew Salmon, 29-Jun-2011.) (Contributed by SF, 5-Aug-1993.) (Revised by SF, 29-Jun-2011.) |
Ref | Expression |
---|---|
pw0 | ⊢ ℘∅ = {∅} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss0b 3580 | . . 3 ⊢ (x ⊆ ∅ ↔ x = ∅) | |
2 | 1 | abbii 2465 | . 2 ⊢ {x ∣ x ⊆ ∅} = {x ∣ x = ∅} |
3 | df-pw 3724 | . 2 ⊢ ℘∅ = {x ∣ x ⊆ ∅} | |
4 | df-sn 3741 | . 2 ⊢ {∅} = {x ∣ x = ∅} | |
5 | 2, 3, 4 | 3eqtr4i 2383 | 1 ⊢ ℘∅ = {∅} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 {cab 2339 ⊆ wss 3257 ∅c0 3550 ℘cpw 3722 {csn 3737 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 |
This theorem is referenced by: pw10 4161 nnpweq 4523 sfin01 4528 |
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