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Mirrors > Home > NFE Home > Th. List > snelpwi | GIF version |
Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.) |
Ref | Expression |
---|---|
snelpwi | ⊢ (A ∈ B → {A} ∈ ℘B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 3852 | . 2 ⊢ (A ∈ B → {A} ⊆ B) | |
2 | snex 4111 | . . 3 ⊢ {A} ∈ V | |
3 | 2 | elpw 3728 | . 2 ⊢ ({A} ∈ ℘B ↔ {A} ⊆ B) |
4 | 1, 3 | sylibr 203 | 1 ⊢ (A ∈ B → {A} ∈ ℘B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 ⊆ wss 3257 ℘cpw 3722 {csn 3737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-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 |
This theorem is referenced by: unipw 4117 |
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