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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 2666 |
. 2
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2 | fo2nd 6007 |
. . . 4
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3 | fofn 5303 |
. . . 4
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4 | 2, 3 | ax-mp 7 |
. . 3
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5 | funfvex 5390 |
. . . 4
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6 | 5 | funfni 5179 |
. . 3
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7 | 4, 6 | mpan 418 |
. 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-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-sbc 2877 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fo 5085 df-fv 5087 df-2nd 5990 |
This theorem is referenced by: elxp7 6019 xpopth 6025 eqop 6026 op1steq 6028 2nd1st 6029 2ndrn 6032 dfoprab3 6040 elopabi 6044 mpofvex 6052 dfmpo 6071 cnvf1olem 6072 cnvoprab 6082 f1od2 6083 xpmapenlem 6693 cnref1o 9335 fsumcnv 11091 qredeu 11617 qdenval 11702 txbas 12262 txdis 12281 |
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