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Mirrors > Home > ILE Home > Th. List > recexgt0 | Unicode version |
Description: Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Ref | Expression |
---|---|
recexgt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-precex 7884 | . 2 | |
2 | 0re 7920 | . . . 4 | |
3 | ltxrlt 7985 | . . . 4 | |
4 | 2, 3 | mpan 422 | . . 3 |
5 | 4 | pm5.32i 451 | . 2 |
6 | ltxrlt 7985 | . . . . 5 | |
7 | 2, 6 | mpan 422 | . . . 4 |
8 | 7 | anbi1d 462 | . . 3 |
9 | 8 | rexbiia 2485 | . 2 |
10 | 1, 5, 9 | 3imtr4i 200 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wrex 2449 class class class wbr 3989 (class class class)co 5853 cr 7773 cc0 7774 c1 7775 cltrr 7778 cmul 7779 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-1re 7868 ax-addrcl 7871 ax-rnegex 7883 ax-precex 7884 |
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 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-ltxr 7959 |
This theorem is referenced by: ltmul1 8511 |
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