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Mirrors > Home > ILE Home > Th. List > 1stexg | GIF version |
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Ref | Expression |
---|---|
1stexg | ⢠(š“ ā š ā (1st āš“) ā V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2750 | . 2 ⢠(š“ ā š ā š“ ā V) | |
2 | fo1st 6160 | . . . 4 ⢠1st :VāontoāV | |
3 | fofn 5442 | . . . 4 ⢠(1st :VāontoāV ā 1st Fn V) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⢠1st Fn V |
5 | funfvex 5534 | . . . 4 ⢠((Fun 1st ā§ š“ ā dom 1st ) ā (1st āš“) ā V) | |
6 | 5 | funfni 5318 | . . 3 ⢠((1st Fn V ā§ š“ ā V) ā (1st āš“) ā V) |
7 | 4, 6 | mpan 424 | . 2 ⢠(š“ ā V ā (1st āš“) ā V) |
8 | 1, 7 | syl 14 | 1 ⢠(š“ ā š ā (1st āš“) ā V) |
Colors of variables: wff set class |
Syntax hints: ā wi 4 ā wcel 2148 Vcvv 2739 Fn wfn 5213 āontoāwfo 5216 ācfv 5218 1st c1st 6141 |
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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
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 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fo 5224 df-fv 5226 df-1st 6143 |
This theorem is referenced by: elxp7 6173 xpopth 6179 eqop 6180 2nd1st 6183 2ndrn 6186 releldm2 6188 reldm 6189 dfoprab3 6194 elopabi 6198 mpofvex 6206 dfmpo 6226 cnvf1olem 6227 cnvoprab 6237 f1od2 6238 disjxp1 6239 xpmapenlem 6851 cnref1o 9652 fsumcnv 11447 fprodcnv 11635 qredeu 12099 qnumval 12187 xpsff1o 12773 txbas 13843 txdis 13862 |
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