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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > frinfm | Structured version Visualization version Unicode version |
Description: A subset of a well-founded set has an infimum. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
frinfm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fri 4801 |
. . . . 5
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2 | 1 | ancom1s 822 |
. . . 4
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3 | 2 | exp43 623 |
. . 3
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4 | 3 | 3imp2 1248 |
. 2
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5 | ssel2 3413 |
. . . . . . . 8
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6 | 5 | adantrr 731 |
. . . . . . 7
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7 | vex 3034 |
. . . . . . . . . . . 12
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8 | vex 3034 |
. . . . . . . . . . . 12
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9 | 7, 8 | brcnv 5022 |
. . . . . . . . . . 11
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10 | 9 | biimpi 199 |
. . . . . . . . . 10
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11 | 10 | con3i 142 |
. . . . . . . . 9
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12 | 11 | ralimi 2796 |
. . . . . . . 8
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13 | 12 | ad2antll 743 |
. . . . . . 7
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14 | breq2 4399 |
. . . . . . . . . . 11
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15 | 14 | rspcev 3136 |
. . . . . . . . . 10
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16 | 15 | ex 441 |
. . . . . . . . 9
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17 | 16 | ralrimivw 2810 |
. . . . . . . 8
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18 | 17 | ad2antrl 742 |
. . . . . . 7
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19 | 6, 13, 18 | jca32 544 |
. . . . . 6
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20 | 19 | ex 441 |
. . . . 5
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21 | 20 | reximdv2 2855 |
. . . 4
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22 | 21 | adantl 473 |
. . 3
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23 | 22 | 3ad2antr2 1196 |
. 2
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24 | 4, 23 | mpd 15 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-br 4396 df-opab 4455 df-fr 4798 df-cnv 4847 |
This theorem is referenced by: welb 32127 |
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