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Mirrors > Home > ILE Home > Th. List > axltwlin | Unicode version |
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltwlin 7899 with ordering on the extended reals. (Contributed by Jim Kingdon, 15-Jan-2020.) |
Ref | Expression |
---|---|
axltwlin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pre-ltwlin 7899 | . 2 | |
2 | ltxrlt 7997 | . . 3 | |
3 | 2 | 3adant3 1017 | . 2 |
4 | ltxrlt 7997 | . . . 4 | |
5 | 4 | 3adant2 1016 | . . 3 |
6 | ltxrlt 7997 | . . . . 5 | |
7 | 6 | ancoms 268 | . . . 4 |
8 | 7 | 3adant1 1015 | . . 3 |
9 | 5, 8 | orbi12d 793 | . 2 |
10 | 1, 3, 9 | 3imtr4d 203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 105 wo 708 w3a 978 wcel 2146 class class class wbr 3998 cr 7785 cltrr 7790 clt 7966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-pre-ltwlin 7899 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-xp 4626 df-pnf 7968 df-mnf 7969 df-ltxr 7971 |
This theorem is referenced by: ltso 8009 letr 8014 lelttr 8020 ltletr 8021 gt0add 8504 reapcotr 8529 sup3exmid 8885 xrltso 9765 rebtwn2zlemstep 10221 expnbnd 10611 leabs 11050 ltabs 11063 abslt 11064 absle 11065 maxabslemlub 11183 suplociccreex 13595 ivthinclemloc 13612 cnplimclemle 13630 reeff1o 13687 efltlemlt 13688 sin0pilem2 13696 coseq0negpitopi 13750 cos02pilt1 13765 |
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