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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2140 |
. . . 4
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2 | orc 702 |
. . . 4
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3 | 1, 2 | mpan2 422 |
. . 3
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4 | 3 | olcd 724 |
. 2
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5 | rexr 7835 |
. . 3
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6 | pnfxr 7842 |
. . 3
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7 | ltxr 9592 |
. . 3
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8 | 5, 6, 7 | sylancl 410 |
. 2
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9 | 4, 8 | mpbird 166 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-cnex 7735 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-xr 7828 df-ltxr 7829 |
This theorem is referenced by: 0ltpnf 9598 xrlttr 9611 xrltso 9612 xrlttri3 9613 nltpnft 9627 npnflt 9628 xrrebnd 9632 xrre 9633 xltnegi 9648 xltadd1 9689 xposdif 9695 elioc2 9749 elicc2 9751 ioomax 9761 ioopos 9763 elioopnf 9780 elicopnf 9782 qbtwnxr 10066 filtinf 10570 xrmaxltsup 11059 xblss2ps 12612 |
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