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Mirrors > Home > ILE Home > Th. List > basmex | Unicode version |
Description: A structure whose base is inhabited is a set. (Contributed by Jim Kingdon, 18-Nov-2024.) |
Ref | Expression |
---|---|
basmex.b |
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Ref | Expression |
---|---|
basmex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | basfn 12499 |
. . . 4
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2 | fnrel 5310 |
. . . 4
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3 | 1, 2 | ax-mp 5 |
. . 3
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4 | basmex.b |
. . . . 5
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5 | 4 | eleq2i 2244 |
. . . 4
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6 | 5 | biimpi 120 |
. . 3
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7 | relelfvdm 5543 |
. . 3
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8 | 3, 6, 7 | sylancr 414 |
. 2
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9 | 8 | elexd 2750 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-cnex 7890 ax-resscn 7891 ax-1re 7893 ax-addrcl 7896 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-inn 8906 df-ndx 12445 df-slot 12446 df-base 12448 |
This theorem is referenced by: ismgmid 12685 ismnd 12709 dfgrp2e 12790 grpinvval 12803 grplactfval 12857 mulgval 12872 mulg1 12876 mulgnnp1 12877 |
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