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Mirrors > Home > ILE Home > Th. List > rrgmex | Unicode version |
Description: A structure whose set of left-regular elements is inhabited is a set. (Contributed by Jim Kingdon, 12-Aug-2025.) |
Ref | Expression |
---|---|
rrgmex.e |
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Ref | Expression |
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rrgmex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptrel 4784 |
. . . 4
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2 | df-rlreg 13738 |
. . . . 5
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3 | 2 | releqi 4738 |
. . . 4
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4 | 1, 3 | mpbir 146 |
. . 3
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5 | rrgmex.e |
. . . . 5
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6 | 5 | eleq2i 2260 |
. . . 4
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7 | 6 | biimpi 120 |
. . 3
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8 | relelfvdm 5578 |
. . 3
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9 | 4, 7, 8 | sylancr 414 |
. 2
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10 | 9 | elexd 2773 |
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-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-xp 4661 df-rel 4662 df-dm 4665 df-iota 5207 df-fv 5254 df-rlreg 13738 |
This theorem is referenced by: rrgval 13742 |
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