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Mirrors > Home > ILE Home > Th. List > xrrebnd | Unicode version |
Description: An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
xrrebnd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9531 | . 2 | |
2 | id 19 | . . . 4 | |
3 | mnflt 9537 | . . . . 5 | |
4 | ltpnf 9535 | . . . . 5 | |
5 | 3, 4 | jca 304 | . . . 4 |
6 | 2, 5 | 2thd 174 | . . 3 |
7 | renepnf 7781 | . . . . 5 | |
8 | 7 | necon2bi 2340 | . . . 4 |
9 | pnfxr 7786 | . . . . . . 7 | |
10 | xrltnr 9534 | . . . . . . 7 | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 |
12 | breq1 3902 | . . . . . 6 | |
13 | 11, 12 | mtbiri 649 | . . . . 5 |
14 | 13 | intnand 901 | . . . 4 |
15 | 8, 14 | 2falsed 676 | . . 3 |
16 | renemnf 7782 | . . . . 5 | |
17 | 16 | necon2bi 2340 | . . . 4 |
18 | mnfxr 7790 | . . . . . . 7 | |
19 | xrltnr 9534 | . . . . . . 7 | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 |
21 | breq2 3903 | . . . . . 6 | |
22 | 20, 21 | mtbiri 649 | . . . . 5 |
23 | 22 | intnanrd 902 | . . . 4 |
24 | 17, 23 | 2falsed 676 | . . 3 |
25 | 6, 15, 24 | 3jaoi 1266 | . 2 |
26 | 1, 25 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 946 wceq 1316 wcel 1465 class class class wbr 3899 cr 7587 cpnf 7765 cmnf 7766 cxr 7767 clt 7768 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 |
This theorem is referenced by: xrre 9571 xrre2 9572 xrre3 9573 elioc2 9687 elico2 9688 elicc2 9689 xblpnfps 12494 xblpnf 12495 |
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