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Mirrors > Home > ILE Home > Th. List > xrpnfdc | Unicode version |
Description: An extended real is or is not plus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.) |
Ref | Expression |
---|---|
xrpnfdc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9563 | . 2 | |
2 | renepnf 7813 | . . . . . 6 | |
3 | 2 | neneqd 2329 | . . . . 5 |
4 | 3 | olcd 723 | . . . 4 |
5 | df-dc 820 | . . . 4 DECID | |
6 | 4, 5 | sylibr 133 | . . 3 DECID |
7 | orc 701 | . . . 4 | |
8 | 7, 5 | sylibr 133 | . . 3 DECID |
9 | mnfnepnf 7821 | . . . . . . 7 | |
10 | 9 | neii 2310 | . . . . . 6 |
11 | eqeq1 2146 | . . . . . 6 | |
12 | 10, 11 | mtbiri 664 | . . . . 5 |
13 | 12 | olcd 723 | . . . 4 |
14 | 13, 5 | sylibr 133 | . . 3 DECID |
15 | 6, 8, 14 | 3jaoi 1281 | . 2 DECID |
16 | 1, 15 | sylbi 120 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 697 DECID wdc 819 w3o 961 wceq 1331 wcel 1480 cr 7619 cpnf 7797 cmnf 7798 cxr 7799 |
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 2121 ax-sep 4046 ax-pow 4098 ax-un 4355 ax-cnex 7711 ax-resscn 7712 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-pnf 7802 df-mnf 7803 df-xr 7804 |
This theorem is referenced by: xaddf 9627 xaddval 9628 xaddpnf1 9629 xaddcom 9644 xnegdi 9651 xleadd1a 9656 xlesubadd 9666 xrmaxiflemcl 11014 xrmaxifle 11015 xrmaxiflemab 11016 xrmaxiflemlub 11017 xrmaxiflemcom 11018 xrmaxadd 11030 xblss2ps 12573 xblss2 12574 |
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