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Mirrors > Home > MPE Home > Th. List > nltpnft | Structured version Visualization version Unicode version |
Description: An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.) |
Ref | Expression |
---|---|
nltpnft |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 11435 |
. . . 4
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2 | xrltnr 11444 |
. . . 4
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3 | 1, 2 | ax-mp 5 |
. . 3
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4 | breq1 4398 |
. . 3
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5 | 3, 4 | mtbiri 310 |
. 2
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6 | pnfge 11455 |
. . . 4
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7 | xrleloe 11466 |
. . . . 5
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8 | 1, 7 | mpan2 685 |
. . . 4
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9 | 6, 8 | mpbid 215 |
. . 3
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10 | 9 | ord 384 |
. 2
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11 | 5, 10 | impbid2 209 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-cnex 9613 ax-resscn 9614 ax-pre-lttri 9631 ax-pre-lttrn 9632 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-po 4760 df-so 4761 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-er 7381 df-en 7588 df-dom 7589 df-sdom 7590 df-pnf 9695 df-mnf 9696 df-xr 9697 df-ltxr 9698 df-le 9699 |
This theorem is referenced by: xrrebnd 11486 xlt2add 11571 supxrbnd1 11632 supxrbnd2 11633 supxrgtmnf 11640 supxrre2 11642 ioopnfsup 12124 icopnfsup 12125 xrsdsreclblem 19091 ovoliun 22536 ovolicopnf 22556 voliunlem3 22584 volsup 22588 itg2seq 22779 nmoreltpnf 26491 nmopreltpnf 27603 xgepnf 28402 ismblfin 32045 supxrgere 37643 supxrgelem 37647 supxrge 37648 suplesup 37649 nepnfltpnf 37652 sge0repnf 38342 sge0rpcpnf 38377 sge0rernmpt 38378 |
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