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Mirrors > Home > ILE Home > Th. List > 2ndexg | GIF version |
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Ref | Expression |
---|---|
2ndexg | ā¢ (š“ ā š ā (2nd āš“) ā V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2748 | . 2 ā¢ (š“ ā š ā š“ ā V) | |
2 | fo2nd 6158 | . . . 4 ā¢ 2nd :VāontoāV | |
3 | fofn 5440 | . . . 4 ā¢ (2nd :VāontoāV ā 2nd Fn V) | |
4 | 2, 3 | ax-mp 5 | . . 3 ā¢ 2nd Fn V |
5 | funfvex 5532 | . . . 4 ā¢ ((Fun 2nd ā§ š“ ā dom 2nd ) ā (2nd āš“) ā V) | |
6 | 5 | funfni 5316 | . . 3 ā¢ ((2nd Fn V ā§ š“ ā V) ā (2nd āš“) ā V) |
7 | 4, 6 | mpan 424 | . 2 ā¢ (š“ ā V ā (2nd āš“) ā V) |
8 | 1, 7 | syl 14 | 1 ā¢ (š“ ā š ā (2nd āš“) ā V) |
Colors of variables: wff set class |
Syntax hints: ā wi 4 ā wcel 2148 Vcvv 2737 Fn wfn 5211 āontoāwfo 5214 ācfv 5216 2nd c2nd 6139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fo 5222 df-fv 5224 df-2nd 6141 |
This theorem is referenced by: elxp7 6170 xpopth 6176 eqop 6177 op1steq 6179 2nd1st 6180 2ndrn 6183 dfoprab3 6191 elopabi 6195 mpofvex 6203 dfmpo 6223 cnvf1olem 6224 cnvoprab 6234 f1od2 6235 xpmapenlem 6848 cc2lem 7264 cnref1o 9649 fsumcnv 11444 fprodcnv 11632 qredeu 12096 qdenval 12185 xpsff1o 12767 txbas 13728 txdis 13747 |
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