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Mirrors > Home > MPE Home > Th. List > cnvex | Structured version Visualization version GIF version |
Description: The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26. (Contributed by NM, 19-Dec-2003.) |
Ref | Expression |
---|---|
cnvex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
cnvex | ⊢ ◡𝐴 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | cnvexg 7702 | . 2 ⊢ (𝐴 ∈ V → ◡𝐴 ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ◡𝐴 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3408 ◡ccnv 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-dm 5561 df-rn 5562 |
This theorem is referenced by: f1oexbi 7706 funcnvuni 7709 cnvf1o 7879 brtpos2 7974 pw2f1o 8750 sbthlem10 8765 fodomr 8797 ssenen 8820 cnfcomlem 9314 infxpenlem 9627 enfin2i 9935 fin1a2lem7 10020 fpwwe 10260 canthwelem 10264 axdc4uzlem 13556 hashfacen 14018 hashfacenOLD 14019 catcisolem 17616 oduleval 17797 gicsubgen 18682 isunit 19675 znle 20501 evpmss 20548 psgnevpmb 20549 ptbasfi 22478 nghmfval 23620 fta1glem2 25064 fta1blem 25066 lgsqrlem4 26230 tocycf 31103 evpmval 31131 altgnsg 31135 elrspunidl 31320 locfinreflem 31504 zarcmplem 31545 qqhval 31636 mbfmcnt 31947 derangenlem 32846 mthmval 33250 colinearex 34099 fvline 34183 ptrest 35513 poimir 35547 tendoi2 38546 dihopelvalcpre 38999 pw2f1ocnv 40562 cnvintabd 40887 clcnvlem 40907 frege133 41281 binomcxplemnotnn0 41647 fzisoeu 42512 isomushgr 44951 isomgrsym 44961 |
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