Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nltpnft | Unicode version |
Description: An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.) |
Ref | Expression |
---|---|
nltpnft |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9706 | . 2 | |
2 | renepnf 7940 | . . . . 5 | |
3 | 2 | neneqd 2355 | . . . 4 |
4 | ltpnf 9710 | . . . . 5 | |
5 | notnot 619 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 3, 6 | 2falsed 692 | . . 3 |
8 | id 19 | . . . 4 | |
9 | pnfxr 7945 | . . . . . 6 | |
10 | xrltnr 9709 | . . . . . 6 | |
11 | 9, 10 | ax-mp 5 | . . . . 5 |
12 | breq1 3982 | . . . . 5 | |
13 | 11, 12 | mtbiri 665 | . . . 4 |
14 | 8, 13 | 2thd 174 | . . 3 |
15 | mnfnepnf 7948 | . . . . . 6 | |
16 | 15 | neii 2336 | . . . . 5 |
17 | eqeq1 2171 | . . . . 5 | |
18 | 16, 17 | mtbiri 665 | . . . 4 |
19 | mnfltpnf 9715 | . . . . . . 7 | |
20 | breq1 3982 | . . . . . . 7 | |
21 | 19, 20 | mpbiri 167 | . . . . . 6 |
22 | 21 | necon3bi 2384 | . . . . 5 |
23 | 22 | necon2bi 2389 | . . . 4 |
24 | 18, 23 | 2falsed 692 | . . 3 |
25 | 7, 14, 24 | 3jaoi 1292 | . 2 |
26 | 1, 25 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 w3o 966 wceq 1342 wcel 2135 class class class wbr 3979 cr 7746 cpnf 7924 cmnf 7925 cxr 7926 clt 7927 |
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 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-pre-ltirr 7859 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 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 2726 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-xp 4607 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 |
This theorem is referenced by: npnflt 9745 xgepnf 9746 xrmaxiflemlub 11183 |
Copyright terms: Public domain | W3C validator |