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Mirrors > Home > ILE Home > Th. List > xrre | Unicode version |
Description: A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.) |
Ref | Expression |
---|---|
xrre |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 520 | . 2 | |
2 | ltpnf 9560 | . . . . . 6 | |
3 | 2 | adantl 275 | . . . . 5 |
4 | rexr 7804 | . . . . . 6 | |
5 | pnfxr 7811 | . . . . . . 7 | |
6 | xrlelttr 9582 | . . . . . . 7 | |
7 | 5, 6 | mp3an3 1304 | . . . . . 6 |
8 | 4, 7 | sylan2 284 | . . . . 5 |
9 | 3, 8 | mpan2d 424 | . . . 4 |
10 | 9 | imp 123 | . . 3 |
11 | 10 | adantrl 469 | . 2 |
12 | xrrebnd 9595 | . . 3 | |
13 | 12 | ad2antrr 479 | . 2 |
14 | 1, 11, 13 | mpbir2and 928 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1480 class class class wbr 3924 cr 7612 cpnf 7790 cmnf 7791 cxr 7792 clt 7793 cle 7794 |
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 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 |
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-po 4213 df-iso 4214 df-xp 4540 df-cnv 4542 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 |
This theorem is referenced by: xrrege0 9601 tgioo 12704 |
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