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Mirrors > Home > MPE Home > Th. List > limsuplt | Structured version Unicode version |
Description: The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.) |
Ref | Expression |
---|---|
limsupval.1 |
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Ref | Expression |
---|---|
limsuplt |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupval.1 |
. . . . 5
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2 | 1 | limsuple 13060 |
. . . 4
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3 | 2 | notbid 294 |
. . 3
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4 | rexnal 2842 |
. . 3
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5 | 3, 4 | syl6bbr 263 |
. 2
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6 | simp2 989 |
. . . . 5
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7 | reex 9476 |
. . . . . . 7
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8 | 7 | ssex 4536 |
. . . . . 6
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9 | 8 | 3ad2ant1 1009 |
. . . . 5
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10 | xrex 11091 |
. . . . . 6
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11 | 10 | a1i 11 |
. . . . 5
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12 | fex2 6634 |
. . . . 5
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13 | 6, 9, 11, 12 | syl3anc 1219 |
. . . 4
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14 | limsupcl 13055 |
. . . 4
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15 | 13, 14 | syl 16 |
. . 3
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16 | simp3 990 |
. . 3
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17 | xrltnle 9546 |
. . 3
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18 | 15, 16, 17 | syl2anc 661 |
. 2
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19 | 1 | limsupgf 13057 |
. . . . . 6
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20 | 19 | ffvelrni 5943 |
. . . . 5
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21 | 20 | adantl 466 |
. . . 4
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22 | simpl3 993 |
. . . 4
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23 | xrltnle 9546 |
. . . 4
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24 | 21, 22, 23 | syl2anc 661 |
. . 3
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25 | 24 | rexbidva 2845 |
. 2
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26 | 5, 18, 25 | 3bitr4d 285 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4513 ax-nul 4521 ax-pow 4570 ax-pr 4631 ax-un 6474 ax-cnex 9441 ax-resscn 9442 ax-1cn 9443 ax-icn 9444 ax-addcl 9445 ax-addrcl 9446 ax-mulcl 9447 ax-mulrcl 9448 ax-mulcom 9449 ax-addass 9450 ax-mulass 9451 ax-distr 9452 ax-i2m1 9453 ax-1ne0 9454 ax-1rid 9455 ax-rnegex 9456 ax-rrecex 9457 ax-cnre 9458 ax-pre-lttri 9459 ax-pre-lttrn 9460 ax-pre-ltadd 9461 ax-pre-mulgt0 9462 ax-pre-sup 9463 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-nul 3738 df-if 3892 df-pw 3962 df-sn 3978 df-pr 3980 df-op 3984 df-uni 4192 df-br 4393 df-opab 4451 df-mpt 4452 df-id 4736 df-po 4741 df-so 4742 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-res 4952 df-ima 4953 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-f1 5523 df-fo 5524 df-f1o 5525 df-fv 5526 df-riota 6153 df-ov 6195 df-oprab 6196 df-mpt2 6197 df-er 7203 df-en 7413 df-dom 7414 df-sdom 7415 df-sup 7794 df-pnf 9523 df-mnf 9524 df-xr 9525 df-ltxr 9526 df-le 9527 df-sub 9700 df-neg 9701 df-limsup 13053 |
This theorem is referenced by: limsupgre 13063 |
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