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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 |
. 2
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2 | fo2nd 6182 |
. . . 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-2nd 6165 |
This theorem is referenced by: elxp7 6194 xpopth 6200 eqop 6201 op1steq 6203 2nd1st 6204 2ndrn 6207 dfoprab3 6215 elopabi 6219 mpofvex 6227 dfmpo 6247 cnvf1olem 6248 cnvoprab 6258 f1od2 6259 xpmapenlem 6876 cc2lem 7294 cnref1o 9679 fsumcnv 11476 fprodcnv 11664 qredeu 12128 qdenval 12217 xpsff1o 12822 txbas 14210 txdis 14229 |
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