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Mirrors > Home > ILE Home > Th. List > axapti | Unicode version |
Description: Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-apti 7876 with ordering on the extended reals.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
Ref | Expression |
---|---|
axapti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltxrlt 7972 | . . . . 5 | |
2 | ltxrlt 7972 | . . . . . 6 | |
3 | 2 | ancoms 266 | . . . . 5 |
4 | 1, 3 | orbi12d 788 | . . . 4 |
5 | 4 | notbid 662 | . . 3 |
6 | ax-pre-apti 7876 | . . . 4 | |
7 | 6 | 3expia 1200 | . . 3 |
8 | 5, 7 | sylbid 149 | . 2 |
9 | 8 | 3impia 1195 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 w3a 973 wceq 1348 wcel 2141 class class class wbr 3987 cr 7760 cltrr 7765 clt 7941 |
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 7852 ax-resscn 7853 ax-pre-apti 7876 |
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-pnf 7943 df-mnf 7944 df-ltxr 7946 |
This theorem is referenced by: lttri3 7986 reapti 8485 |
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