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Mirrors > Home > ILE Home > Th. List > 2ndexg | Unicode version |
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Ref | Expression |
---|---|
2ndexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2741 | . 2 | |
2 | fo2nd 6134 | . . . 4 | |
3 | fofn 5420 | . . . 4 | |
4 | 2, 3 | ax-mp 5 | . . 3 |
5 | funfvex 5511 | . . . 4 | |
6 | 5 | funfni 5296 | . . 3 |
7 | 4, 6 | mpan 422 | . 2 |
8 | 1, 7 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2141 cvv 2730 wfn 5191 wfo 5194 cfv 5196 c2nd 6115 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fo 5202 df-fv 5204 df-2nd 6117 |
This theorem is referenced by: elxp7 6146 xpopth 6152 eqop 6153 op1steq 6155 2nd1st 6156 2ndrn 6159 dfoprab3 6167 elopabi 6171 mpofvex 6179 dfmpo 6199 cnvf1olem 6200 cnvoprab 6210 f1od2 6211 xpmapenlem 6823 cc2lem 7215 cnref1o 9596 fsumcnv 11387 fprodcnv 11575 qredeu 12038 qdenval 12127 txbas 13011 txdis 13030 |
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