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Mirrors > Home > ILE Home > Th. List > infglbti | Unicode version |
Description: An infimum is the greatest lower bound. See also infclti 7036 and inflbti 7037. (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 6998 |
. . . . 5
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2 | 1 | breq1i 4022 |
. . . 4
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3 | simpr 110 |
. . . . 5
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4 | infclti.ti |
. . . . . . . 8
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5 | 4 | cnvti 7032 |
. . . . . . 7
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6 | infclti.ex |
. . . . . . . 8
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7 | 6 | cnvinfex 7031 |
. . . . . . 7
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8 | 5, 7 | supclti 7011 |
. . . . . 6
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9 | 8 | adantr 276 |
. . . . 5
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10 | brcnvg 4820 |
. . . . . 6
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11 | 10 | bicomd 141 |
. . . . 5
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12 | 3, 9, 11 | syl2anc 411 |
. . . 4
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13 | 2, 12 | bitrid 192 |
. . 3
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14 | 5, 7 | suplubti 7013 |
. . . . 5
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15 | 14 | expdimp 259 |
. . . 4
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16 | vex 2752 |
. . . . . 6
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17 | brcnvg 4820 |
. . . . . 6
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18 | 3, 16, 17 | sylancl 413 |
. . . . 5
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19 | 18 | rexbidv 2488 |
. . . 4
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20 | 15, 19 | sylibd 149 |
. . 3
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21 | 13, 20 | sylbid 150 |
. 2
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22 | 21 | expimpd 363 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-cnv 4646 df-iota 5190 df-riota 5844 df-sup 6997 df-inf 6998 |
This theorem is referenced by: infnlbti 7039 zssinfcl 11963 |
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