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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 7937 | . 2 ⊢ (𝐴 ∈ V → ◡𝐴 ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ◡𝐴 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 Vcvv 3462 ◡ccnv 5681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 |
This theorem is referenced by: f1oexbi 7941 funcnvuni 7945 cnvf1o 8125 brtpos2 8247 pw2f1o 9115 sbthlem10 9130 fodomr 9166 ssenen 9189 cnfcomlem 9742 infxpenlem 10056 enfin2i 10364 fin1a2lem7 10449 fpwwe 10689 canthwelem 10693 axdc4uzlem 14003 hashfacen 14471 hashfacenOLD 14472 catcisolem 18132 oduleval 18314 gicsubgen 19273 isunit 20355 znle 21530 evpmss 21582 psgnevpmb 21583 ptbasfi 23576 nghmfval 24730 fta1glem2 26196 fta1blem 26198 lgsqrlem4 27378 tocycf 32995 evpmval 33023 altgnsg 33027 elrspunidl 33303 1arithidom 33412 irngval 33561 locfinreflem 33655 zarcmplem 33696 qqhval 33789 mbfmcnt 34102 derangenlem 34999 mthmval 35403 colinearex 35884 fvline 35968 ptrest 37320 poimir 37354 tendoi2 40494 dihopelvalcpre 40947 pw2f1ocnv 42695 cnvintabd 43270 clcnvlem 43290 frege133 43663 binomcxplemnotnn0 44030 fzisoeu 44915 gricushgr 47465 |
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