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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 |
. 2
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2 | fo1st 6183 |
. . . 4
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3 | fofn 5459 |
. . . 4
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4 | 2, 3 | ax-mp 5 |
. . 3
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5 | funfvex 5551 |
. . . 4
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6 | 5 | funfni 5335 |
. . 3
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7 | 4, 6 | mpan 424 |
. 2
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8 | 1, 7 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fo 5241 df-fv 5243 df-1st 6166 |
This theorem is referenced by: elxp7 6196 xpopth 6202 eqop 6203 2nd1st 6206 2ndrn 6209 releldm2 6211 reldm 6212 dfoprab3 6217 elopabi 6221 mpofvex 6229 dfmpo 6249 cnvf1olem 6250 cnvoprab 6260 f1od2 6261 disjxp1 6262 xpmapenlem 6878 cnref1o 9682 fsumcnv 11480 fprodcnv 11668 qredeu 12132 qnumval 12220 xpsff1o 12828 txbas 14235 txdis 14254 |
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