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Mirrors > Home > NFE Home > Th. List > complex | GIF version |
Description: The complement of a set is a set. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
boolex.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
complex | ⊢ ∼ A ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | boolex.1 | . 2 ⊢ A ∈ V | |
2 | complexg 4099 | . 2 ⊢ (A ∈ V → ∼ A ∈ V) | |
3 | 1, 2 | ax-mp 8 | 1 ⊢ ∼ A ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 Vcvv 2859 ∼ ccompl 3205 |
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 |
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 |
This theorem is referenced by: vvex 4109 0ex 4110 imakexg 4299 intexg 4319 addcexlem 4382 nnsucelrlem1 4424 nndisjeq 4429 preaddccan2lem1 4454 ltfinex 4464 ssfin 4470 ncfinraiselem2 4480 ncfinlowerlem1 4482 tfinrelkex 4487 evenfinex 4503 oddfinex 4504 evenodddisjlem1 4515 nnadjoinlem1 4519 nnpweqlem1 4522 srelkex 4525 sfintfinlem1 4531 tfinnnlem1 4533 spfinex 4537 vfintle 4546 vfin1cltv 4547 nulnnn 4556 phiexg 4571 opexg 4587 proj1exg 4591 proj2exg 4592 setconslem5 4735 1stex 4739 swapex 4742 nfunv 5138 mptexlem 5810 disjex 5823 funsex 5828 fullfunexg 5859 transex 5910 refex 5911 antisymex 5912 connexex 5913 foundex 5914 extex 5915 symex 5916 endisj 6051 enprmaplem4 6079 nenpw1pwlem1 6084 ncaddccl 6144 tcdi 6164 ovcelem1 6171 ceex 6174 ce0nn 6180 tcfnex 6244 nclennlem1 6248 nmembers1lem1 6268 nnc3n3p1 6278 nchoicelem11 6299 nchoicelem16 6304 nchoicelem18 6306 |
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