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Mirrors > Home > NFE Home > Th. List > pwexg | GIF version |
Description: The power class of a set is a set. (Contributed by SF, 21-Jan-2015.) |
Ref | Expression |
---|---|
pwexg | ⊢ (A ∈ V → ℘A ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpw2 4327 | . 2 ⊢ ℘A = ∼ (( Sk ∖ (℘1A ×k V)) “k 1c) | |
2 | ssetkex 4294 | . . . . 5 ⊢ Sk ∈ V | |
3 | pw1exg 4302 | . . . . . 6 ⊢ (A ∈ V → ℘1A ∈ V) | |
4 | vvex 4109 | . . . . . 6 ⊢ V ∈ V | |
5 | xpkexg 4288 | . . . . . 6 ⊢ ((℘1A ∈ V ∧ V ∈ V) → (℘1A ×k V) ∈ V) | |
6 | 3, 4, 5 | sylancl 643 | . . . . 5 ⊢ (A ∈ V → (℘1A ×k V) ∈ V) |
7 | difexg 4102 | . . . . 5 ⊢ (( Sk ∈ V ∧ (℘1A ×k V) ∈ V) → ( Sk ∖ (℘1A ×k V)) ∈ V) | |
8 | 2, 6, 7 | sylancr 644 | . . . 4 ⊢ (A ∈ V → ( Sk ∖ (℘1A ×k V)) ∈ V) |
9 | 1cex 4142 | . . . 4 ⊢ 1c ∈ V | |
10 | imakexg 4299 | . . . 4 ⊢ ((( Sk ∖ (℘1A ×k V)) ∈ V ∧ 1c ∈ V) → (( Sk ∖ (℘1A ×k V)) “k 1c) ∈ V) | |
11 | 8, 9, 10 | sylancl 643 | . . 3 ⊢ (A ∈ V → (( Sk ∖ (℘1A ×k V)) “k 1c) ∈ V) |
12 | complexg 4099 | . . 3 ⊢ ((( Sk ∖ (℘1A ×k V)) “k 1c) ∈ V → ∼ (( Sk ∖ (℘1A ×k V)) “k 1c) ∈ V) | |
13 | 11, 12 | syl 15 | . 2 ⊢ (A ∈ V → ∼ (( Sk ∖ (℘1A ×k V)) “k 1c) ∈ V) |
14 | 1, 13 | syl5eqel 2437 | 1 ⊢ (A ∈ V → ℘A ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 Vcvv 2859 ∼ ccompl 3205 ∖ cdif 3206 ℘cpw 3722 1cc1c 4134 ℘1cpw1 4135 ×k cxpk 4174 “k cimak 4179 Sk cssetk 4183 |
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-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 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-pr 3742 df-opk 4058 df-1c 4136 df-pw1 4137 df-xpk 4185 df-cnvk 4186 df-imak 4189 df-p6 4191 df-sik 4192 df-ssetk 4193 |
This theorem is referenced by: pwex 4329 pmex 6005 ltcpw1pwg 6202 |
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