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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0neqmnf | Structured version Unicode version |
Description: An extended nonnegative real cannot be minus infinity. (Contributed by Thierry Arnoux, 9-Jun-2017.) |
Ref | Expression |
---|---|
xrge0neqmnf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnflt0 11209 |
. . . . 5
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2 | mnfxr 11198 |
. . . . . 6
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3 | 0xr 9534 |
. . . . . 6
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4 | xrltnle 9547 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 2, 3, 4 | mp2an 672 |
. . . . 5
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6 | 1, 5 | mpbi 208 |
. . . 4
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7 | simp2 989 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 7 | con3i 135 |
. . . 4
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9 | pnfxr 11196 |
. . . . . . 7
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10 | elicc1 11448 |
. . . . . . 7
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11 | 3, 9, 10 | mp2an 672 |
. . . . . 6
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12 | 11 | biimpi 194 |
. . . . 5
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13 | 12 | con3i 135 |
. . . 4
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14 | 6, 8, 13 | mp2b 10 |
. . 3
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15 | nelneq 2568 |
. . 3
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16 | 14, 15 | mpan2 671 |
. 2
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17 | 16 | neneqad 2652 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475 ax-cnex 9442 ax-resscn 9443 ax-1cn 9444 ax-icn 9445 ax-addcl 9446 ax-addrcl 9447 ax-mulcl 9448 ax-mulrcl 9449 ax-i2m1 9454 ax-1ne0 9455 ax-rnegex 9457 ax-rrecex 9458 ax-cnre 9459 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3073 df-sbc 3288 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-br 4394 df-opab 4452 df-id 4737 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-iota 5482 df-fun 5521 df-fv 5527 df-ov 6196 df-oprab 6197 df-mpt2 6198 df-pnf 9524 df-mnf 9525 df-xr 9526 df-ltxr 9527 df-le 9528 df-icc 11411 |
This theorem is referenced by: xrge0nre 26291 xrge0adddir 26293 xrge0npcan 26295 hasheuni 26672 |
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