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Mirrors > Home > MPE Home > Th. List > supxrunb2 | Structured version Visualization version Unicode version |
Description: The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.) |
Ref | Expression |
---|---|
supxrunb2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3428 |
. . . . . . . 8
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2 | pnfnlt 11437 |
. . . . . . . 8
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3 | 1, 2 | syl6 34 |
. . . . . . 7
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4 | 3 | ralrimiv 2802 |
. . . . . 6
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5 | 4 | adantr 467 |
. . . . 5
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6 | breq1 4408 |
. . . . . . . . . . . . . . 15
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7 | 6 | rexbidv 2903 |
. . . . . . . . . . . . . 14
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8 | 7 | rspcva 3150 |
. . . . . . . . . . . . 13
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9 | 8 | adantrr 724 |
. . . . . . . . . . . 12
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10 | 9 | ancoms 455 |
. . . . . . . . . . 11
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11 | 10 | exp31 609 |
. . . . . . . . . 10
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12 | 11 | a1dd 47 |
. . . . . . . . 9
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13 | 12 | com4r 89 |
. . . . . . . 8
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14 | 13 | com13 83 |
. . . . . . 7
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15 | 14 | imp 431 |
. . . . . 6
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16 | 15 | ralrimiv 2802 |
. . . . 5
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17 | 5, 16 | jca 535 |
. . . 4
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18 | pnfxr 11419 |
. . . . 5
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19 | supxr 11605 |
. . . . 5
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20 | 18, 19 | mpanl2 688 |
. . . 4
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21 | 17, 20 | syldan 473 |
. . 3
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22 | 21 | ex 436 |
. 2
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23 | rexr 9691 |
. . . . . . 7
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24 | 23 | ad2antlr 734 |
. . . . . 6
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25 | ltpnf 11429 |
. . . . . . . . 9
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26 | breq2 4409 |
. . . . . . . . 9
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27 | 25, 26 | syl5ibr 225 |
. . . . . . . 8
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28 | 27 | impcom 432 |
. . . . . . 7
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29 | 28 | adantll 721 |
. . . . . 6
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30 | xrltso 11447 |
. . . . . . . 8
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31 | 30 | a1i 11 |
. . . . . . 7
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32 | xrsupss 11601 |
. . . . . . . 8
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33 | 32 | ad2antrr 733 |
. . . . . . 7
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34 | 31, 33 | suplub 7979 |
. . . . . 6
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35 | 24, 29, 34 | mp2and 686 |
. . . . 5
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36 | 35 | exp31 609 |
. . . 4
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37 | 36 | com23 81 |
. . 3
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38 | 37 | ralrimdv 2806 |
. 2
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39 | 22, 38 | impbid 194 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 ax-pre-mulgt0 9621 ax-pre-sup 9622 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 df-po 4758 df-so 4759 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-er 7368 df-en 7575 df-dom 7576 df-sdom 7577 df-sup 7961 df-pnf 9682 df-mnf 9683 df-xr 9684 df-ltxr 9685 df-le 9686 df-sub 9867 df-neg 9868 |
This theorem is referenced by: supxrbnd2 11615 supxrbnd 11621 suplesup 37572 sge0pnffigt 38248 |
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