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Mirrors > Home > ILE Home > Th. List > infglbti | Unicode version |
Description: An infimum is the greatest lower bound. See also infclti 6862 and inflbti 6863. (Contributed by Jim Kingdon, 18-Dec-2021.) |
Ref | Expression |
---|---|
infclti.ti |
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infclti.ex |
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Ref | Expression |
---|---|
infglbti |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6824 |
. . . . 5
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2 | 1 | breq1i 3902 |
. . . 4
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3 | simpr 109 |
. . . . 5
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4 | infclti.ti |
. . . . . . . 8
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5 | 4 | cnvti 6858 |
. . . . . . 7
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6 | infclti.ex |
. . . . . . . 8
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7 | 6 | cnvinfex 6857 |
. . . . . . 7
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8 | 5, 7 | supclti 6837 |
. . . . . 6
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9 | 8 | adantr 272 |
. . . . 5
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10 | brcnvg 4680 |
. . . . . 6
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11 | 10 | bicomd 140 |
. . . . 5
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12 | 3, 9, 11 | syl2anc 406 |
. . . 4
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13 | 2, 12 | syl5bb 191 |
. . 3
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14 | 5, 7 | suplubti 6839 |
. . . . 5
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15 | 14 | expdimp 257 |
. . . 4
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16 | vex 2660 |
. . . . . 6
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17 | brcnvg 4680 |
. . . . . 6
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18 | 3, 16, 17 | sylancl 407 |
. . . . 5
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19 | 18 | rexbidv 2412 |
. . . 4
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20 | 15, 19 | sylibd 148 |
. . 3
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21 | 13, 20 | sylbid 149 |
. 2
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22 | 21 | expimpd 358 |
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-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-reu 2397 df-rmo 2398 df-rab 2399 df-v 2659 df-sbc 2879 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-cnv 4507 df-iota 5046 df-riota 5684 df-sup 6823 df-inf 6824 |
This theorem is referenced by: infnlbti 6865 zssinfcl 11489 |
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