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Mirrors > Home > ILE Home > Th. List > supsnti | Unicode version |
Description: The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.) |
Ref | Expression |
---|---|
supsnti.ti |
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supsnti.b |
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Ref | Expression |
---|---|
supsnti |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supsnti.ti |
. 2
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2 | supsnti.b |
. 2
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3 | snidg 3636 |
. . 3
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4 | 2, 3 | syl 14 |
. 2
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5 | eqid 2189 |
. . . . . 6
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6 | 1 | ralrimivva 2572 |
. . . . . . 7
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7 | eqeq1 2196 |
. . . . . . . . . 10
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8 | breq1 4021 |
. . . . . . . . . . . 12
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9 | 8 | notbid 668 |
. . . . . . . . . . 11
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10 | breq2 4022 |
. . . . . . . . . . . 12
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11 | 10 | notbid 668 |
. . . . . . . . . . 11
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12 | 9, 11 | anbi12d 473 |
. . . . . . . . . 10
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13 | 7, 12 | bibi12d 235 |
. . . . . . . . 9
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14 | eqeq2 2199 |
. . . . . . . . . 10
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15 | breq2 4022 |
. . . . . . . . . . . 12
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16 | 15 | notbid 668 |
. . . . . . . . . . 11
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17 | breq1 4021 |
. . . . . . . . . . . 12
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18 | 17 | notbid 668 |
. . . . . . . . . . 11
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19 | 16, 18 | anbi12d 473 |
. . . . . . . . . 10
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20 | 14, 19 | bibi12d 235 |
. . . . . . . . 9
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21 | 13, 20 | rspc2v 2869 |
. . . . . . . 8
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22 | 2, 2, 21 | syl2anc 411 |
. . . . . . 7
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23 | 6, 22 | mpd 13 |
. . . . . 6
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24 | 5, 23 | mpbii 148 |
. . . . 5
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25 | 24 | simpld 112 |
. . . 4
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26 | 25 | adantr 276 |
. . 3
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27 | elsni 3625 |
. . . . . 6
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28 | 27 | breq2d 4030 |
. . . . 5
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29 | 28 | notbid 668 |
. . . 4
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30 | 29 | adantl 277 |
. . 3
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31 | 26, 30 | mpbird 167 |
. 2
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32 | 1, 2, 4, 31 | supmaxti 7032 |
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-in1 615 ax-in2 616 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-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 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-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5196 df-riota 5851 df-sup 7012 |
This theorem is referenced by: infsnti 7058 |
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