Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7112 | . 2 ⊢ (𝐴 ∈ V → ◡𝐴 ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ◡𝐴 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 Vcvv 3200 ◡ccnv 5113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: f1oexbi 7116 funcnvuni 7119 cnvf1o 7276 brtpos2 7358 pw2f1o 8065 sbthlem10 8079 fodomr 8111 ssenen 8134 cnfcomlem 8596 infxpenlem 8836 enfin2i 9143 fin1a2lem7 9228 fpwwe 9468 canthwelem 9472 axdc4uzlem 12782 hashfacen 13238 xpscf 16226 xpsfval 16227 xpssca 16238 xpsvsca 16239 catcisolem 16756 oduleval 17131 gicsubgen 17721 isunit 18657 znle 19884 evpmss 19932 psgnevpmb 19933 ptbasfi 21384 nghmfval 22526 fta1glem2 23926 fta1blem 23928 lgsqrlem4 25074 locfinreflem 29907 qqhval 30018 mbfmcnt 30330 derangenlem 31153 mthmval 31472 colinearex 32167 fvline 32251 ptrest 33408 poimir 33442 tendoi2 36083 dihopelvalcpre 36537 pw2f1ocnv 37604 cnvintabd 37909 clcnvlem 37930 frege133 38290 binomcxplemnotnn0 38555 fzisoeu 39514 |
Copyright terms: Public domain | W3C validator |