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Mirrors > Home > ILE Home > Th. List > npnflt | Unicode version |
Description: An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.) |
Ref | Expression |
---|---|
npnflt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nltpnft 9590 | . . . 4 | |
2 | 1 | biimpd 143 | . . 3 |
3 | 2 | necon2ad 2363 | . 2 |
4 | ltpnf 9560 | . . . . 5 | |
5 | 4 | adantl 275 | . . . 4 |
6 | simpr 109 | . . . . 5 | |
7 | simplr 519 | . . . . 5 | |
8 | 6, 7 | pm2.21ddne 2389 | . . . 4 |
9 | mnfltpnf 9564 | . . . . . 6 | |
10 | breq1 3927 | . . . . . 6 | |
11 | 9, 10 | mpbiri 167 | . . . . 5 |
12 | 11 | adantl 275 | . . . 4 |
13 | elxr 9556 | . . . . . 6 | |
14 | 13 | biimpi 119 | . . . . 5 |
15 | 14 | adantr 274 | . . . 4 |
16 | 5, 8, 12, 15 | mpjao3dan 1285 | . . 3 |
17 | 16 | ex 114 | . 2 |
18 | 3, 17 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 961 wceq 1331 wcel 1480 wne 2306 class class class wbr 3924 cr 7612 cpnf 7790 cmnf 7791 cxr 7792 clt 7793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 |
This theorem is referenced by: xlt2add 9656 xrmaxadd 11023 |
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