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Mirrors > Home > ILE Home > Th. List > ltpnf | Unicode version |
Description: Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
ltpnf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2040 | . . . 4 | |
2 | orc 633 | . . . 4 | |
3 | 1, 2 | mpan2 401 | . . 3 |
4 | 3 | olcd 653 | . 2 |
5 | rexr 7071 | . . 3 | |
6 | pnfxr 8692 | . . 3 | |
7 | ltxr 8695 | . . 3 | |
8 | 5, 6, 7 | sylancl 392 | . 2 |
9 | 4, 8 | mpbird 156 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wo 629 wceq 1243 wcel 1393 class class class wbr 3764 cr 6888 cltrr 6893 cpnf 7057 cmnf 7058 cxr 7059 clt 7060 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-cnex 6975 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-xr 7064 df-ltxr 7065 |
This theorem is referenced by: 0ltpnf 8703 xrlttr 8716 xrltso 8717 xrlttri3 8718 nltpnft 8730 xrrebnd 8732 xrre 8733 xltnegi 8748 elioc2 8805 elicc2 8807 ioomax 8817 ioopos 8819 elioopnf 8836 elicopnf 8838 |
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