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Mirrors > Home > ILE Home > Th. List > ltrelxr | Unicode version |
Description: 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
ltrelxr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ltxr 6862 |
. 2
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2 | df-3an 886 |
. . . . . 6
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3 | 2 | opabbii 3815 |
. . . . 5
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4 | opabssxp 4357 |
. . . . 5
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5 | 3, 4 | eqsstri 2969 |
. . . 4
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6 | rexpssxrxp 6867 |
. . . 4
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7 | 5, 6 | sstri 2948 |
. . 3
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8 | ressxr 6866 |
. . . . . 6
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9 | snsspr2 3504 |
. . . . . . 7
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10 | ssun2 3101 |
. . . . . . . 8
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11 | df-xr 6861 |
. . . . . . . 8
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12 | 10, 11 | sseqtr4i 2972 |
. . . . . . 7
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13 | 9, 12 | sstri 2948 |
. . . . . 6
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14 | 8, 13 | unssi 3112 |
. . . . 5
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15 | snsspr1 3503 |
. . . . . 6
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16 | 15, 12 | sstri 2948 |
. . . . 5
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17 | xpss12 4388 |
. . . . 5
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18 | 14, 16, 17 | mp2an 402 |
. . . 4
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19 | xpss12 4388 |
. . . . 5
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20 | 13, 8, 19 | mp2an 402 |
. . . 4
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21 | 18, 20 | unssi 3112 |
. . 3
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22 | 7, 21 | unssi 3112 |
. 2
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23 | 1, 22 | eqsstri 2969 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pr 3374 df-opab 3810 df-xp 4294 df-xr 6861 df-ltxr 6862 |
This theorem is referenced by: ltrel 6878 |
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