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Mirrors > Home > ILE Home > Th. List > squeeze0 | Unicode version |
Description: If a nonnegative number is less than any positive number, it is zero. (Contributed by NM, 11-Feb-2006.) |
Ref | Expression |
---|---|
squeeze0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnr 7989 | . . . . 5 | |
2 | 1 | 3ad2ant1 1013 | . . . 4 |
3 | breq2 3991 | . . . . . . 7 | |
4 | breq2 3991 | . . . . . . 7 | |
5 | 3, 4 | imbi12d 233 | . . . . . 6 |
6 | 5 | rspcva 2832 | . . . . 5 |
7 | 6 | 3adant2 1011 | . . . 4 |
8 | 2, 7 | mtod 658 | . . 3 |
9 | simp1 992 | . . . 4 | |
10 | 0red 7914 | . . . 4 | |
11 | 9, 10 | lenltd 8030 | . . 3 |
12 | 8, 11 | mpbird 166 | . 2 |
13 | simp2 993 | . 2 | |
14 | 9, 10 | letri3d 8028 | . 2 |
15 | 12, 13, 14 | mpbir2and 939 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 w3a 973 wceq 1348 wcel 2141 wral 2448 class class class wbr 3987 cr 7766 cc0 7767 clt 7947 cle 7948 |
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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7858 ax-resscn 7859 ax-1re 7861 ax-addrcl 7864 ax-rnegex 7876 ax-pre-ltirr 7879 ax-pre-apti 7882 |
This theorem depends on definitions: df-bi 116 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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-cnv 4617 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 |
This theorem is referenced by: (None) |
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