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Mirrors > Home > ILE Home > Th. List > ngtmnft | Unicode version |
Description: An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
ngtmnft |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9733 | . 2 | |
2 | renemnf 7968 | . . . . 5 | |
3 | 2 | neneqd 2361 | . . . 4 |
4 | mnflt 9740 | . . . . 5 | |
5 | notnot 624 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 3, 6 | 2falsed 697 | . . 3 |
8 | pnfnemnf 7974 | . . . . . 6 | |
9 | neeq1 2353 | . . . . . 6 | |
10 | 8, 9 | mpbiri 167 | . . . . 5 |
11 | 10 | neneqd 2361 | . . . 4 |
12 | mnfltpnf 9742 | . . . . . . 7 | |
13 | breq2 3993 | . . . . . . 7 | |
14 | 12, 13 | mpbiri 167 | . . . . . 6 |
15 | 14 | necon3bi 2390 | . . . . 5 |
16 | 15 | necon2bi 2395 | . . . 4 |
17 | 11, 16 | 2falsed 697 | . . 3 |
18 | id 19 | . . . 4 | |
19 | mnfxr 7976 | . . . . . 6 | |
20 | xrltnr 9736 | . . . . . 6 | |
21 | 19, 20 | ax-mp 5 | . . . . 5 |
22 | breq2 3993 | . . . . 5 | |
23 | 21, 22 | mtbiri 670 | . . . 4 |
24 | 18, 23 | 2thd 174 | . . 3 |
25 | 7, 17, 24 | 3jaoi 1298 | . 2 |
26 | 1, 25 | sylbi 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 w3o 972 wceq 1348 wcel 2141 wne 2340 class class class wbr 3989 cr 7773 cpnf 7951 cmnf 7952 cxr 7953 clt 7954 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltirr 7886 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 |
This theorem is referenced by: nmnfgt 9775 ge0nemnf 9781 xleaddadd 9844 |
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