Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 1stexg | Unicode version |
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Ref | Expression |
---|---|
1stexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2692 | . 2 | |
2 | fo1st 6048 | . . . 4 | |
3 | fofn 5342 | . . . 4 | |
4 | 2, 3 | ax-mp 5 | . . 3 |
5 | funfvex 5431 | . . . 4 | |
6 | 5 | funfni 5218 | . . 3 |
7 | 4, 6 | mpan 420 | . 2 |
8 | 1, 7 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 cvv 2681 wfn 5113 wfo 5116 cfv 5118 c1st 6029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fo 5124 df-fv 5126 df-1st 6031 |
This theorem is referenced by: elxp7 6061 xpopth 6067 eqop 6068 2nd1st 6071 2ndrn 6074 releldm2 6076 reldm 6077 dfoprab3 6082 elopabi 6086 mpofvex 6094 dfmpo 6113 cnvf1olem 6114 cnvoprab 6124 f1od2 6125 disjxp1 6126 xpmapenlem 6736 cnref1o 9433 fsumcnv 11199 qredeu 11767 qnumval 11852 txbas 12416 txdis 12435 |
Copyright terms: Public domain | W3C validator |