Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xrmnfdc | Unicode version |
Description: An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.) |
Ref | Expression |
---|---|
xrmnfdc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9703 | . 2 | |
2 | renemnf 7938 | . . . . . 6 | |
3 | 2 | neneqd 2355 | . . . . 5 |
4 | 3 | olcd 724 | . . . 4 |
5 | df-dc 825 | . . . 4 DECID | |
6 | 4, 5 | sylibr 133 | . . 3 DECID |
7 | pnfnemnf 7944 | . . . . . . 7 | |
8 | 7 | neii 2336 | . . . . . 6 |
9 | eqeq1 2171 | . . . . . 6 | |
10 | 8, 9 | mtbiri 665 | . . . . 5 |
11 | 10 | olcd 724 | . . . 4 |
12 | 11, 5 | sylibr 133 | . . 3 DECID |
13 | orc 702 | . . . 4 | |
14 | 13, 5 | sylibr 133 | . . 3 DECID |
15 | 6, 12, 14 | 3jaoi 1292 | . 2 DECID |
16 | 1, 15 | sylbi 120 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 698 DECID wdc 824 w3o 966 wceq 1342 wcel 2135 cr 7743 cpnf 7921 cmnf 7922 cxr 7923 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-pnf 7926 df-mnf 7927 df-xr 7928 |
This theorem is referenced by: xaddf 9771 xaddval 9772 xaddmnf1 9775 xaddcom 9788 xnegdi 9795 xpncan 9798 xleadd1a 9800 xsubge0 9808 xrmaxiflemcl 11172 xrmaxifle 11173 xrmaxiflemab 11174 xrmaxiflemlub 11175 xrmaxiflemcom 11176 xrmaxadd 11188 |
Copyright terms: Public domain | W3C validator |