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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 7899 | . 2 ⊢ (𝐴 ∈ V → ◡𝐴 ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ◡𝐴 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 Vcvv 3474 ◡ccnv 5669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-xp 5676 df-rel 5677 df-cnv 5678 df-dm 5680 df-rn 5681 |
This theorem is referenced by: f1oexbi 7903 funcnvuni 7906 cnvf1o 8081 brtpos2 8201 pw2f1o 9062 sbthlem10 9077 fodomr 9113 ssenen 9136 cnfcomlem 9678 infxpenlem 9992 enfin2i 10300 fin1a2lem7 10385 fpwwe 10625 canthwelem 10629 axdc4uzlem 13932 hashfacen 14397 hashfacenOLD 14398 catcisolem 18044 oduleval 18226 gicsubgen 19120 isunit 20141 znle 21023 evpmss 21074 psgnevpmb 21075 ptbasfi 23016 nghmfval 24170 fta1glem2 25615 fta1blem 25617 lgsqrlem4 26781 tocycf 32211 evpmval 32239 altgnsg 32243 elrspunidl 32461 irngval 32651 locfinreflem 32715 zarcmplem 32756 qqhval 32849 mbfmcnt 33162 derangenlem 34057 mthmval 34461 colinearex 34926 fvline 35010 ptrest 36355 poimir 36389 tendoi2 39535 dihopelvalcpre 39988 pw2f1ocnv 41611 cnvintabd 42189 clcnvlem 42209 frege133 42582 binomcxplemnotnn0 42950 fzisoeu 43847 isomushgr 46330 isomgrsym 46340 |
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