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Mirrors > Home > NFE Home > Th. List > pw10b | GIF version |
Description: The unit power class of a class is empty iff the class itself is empty. (Contributed by SF, 22-Jan-2015.) |
Ref | Expression |
---|---|
pw10b | ⊢ (℘1A = ∅ ↔ A = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3559 | . . . 4 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A) | |
2 | snelpw1 4146 | . . . . . 6 ⊢ ({x} ∈ ℘1A ↔ x ∈ A) | |
3 | ne0i 3556 | . . . . . 6 ⊢ ({x} ∈ ℘1A → ℘1A ≠ ∅) | |
4 | 2, 3 | sylbir 204 | . . . . 5 ⊢ (x ∈ A → ℘1A ≠ ∅) |
5 | 4 | exlimiv 1634 | . . . 4 ⊢ (∃x x ∈ A → ℘1A ≠ ∅) |
6 | 1, 5 | sylbi 187 | . . 3 ⊢ (A ≠ ∅ → ℘1A ≠ ∅) |
7 | 6 | necon4i 2576 | . 2 ⊢ (℘1A = ∅ → A = ∅) |
8 | pw1eq 4143 | . . 3 ⊢ (A = ∅ → ℘1A = ℘1∅) | |
9 | pw10 4161 | . . 3 ⊢ ℘1∅ = ∅ | |
10 | 8, 9 | syl6eq 2401 | . 2 ⊢ (A = ∅ → ℘1A = ∅) |
11 | 7, 10 | impbii 180 | 1 ⊢ (℘1A = ∅ ↔ A = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∅c0 3550 {csn 3737 ℘1cpw1 4135 |
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 ax-nin 4078 ax-sn 4087 |
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-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-1c 4136 df-pw1 4137 |
This theorem is referenced by: ncfinlower 4483 |
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