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Mirrors > Home > MPE Home > Th. List > nlt1pi | Structured version Visualization version Unicode version |
Description: No positive integer is less than one. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nlt1pi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elni 9301 |
. . . 4
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2 | 1 | simprbi 466 |
. . 3
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3 | noel 3735 |
. . . . . 6
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4 | 1pi 9308 |
. . . . . . . . . 10
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5 | ltpiord 9312 |
. . . . . . . . . 10
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6 | 4, 5 | mpan2 677 |
. . . . . . . . 9
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7 | df-1o 7182 |
. . . . . . . . . . 11
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8 | 7 | eleq2i 2521 |
. . . . . . . . . 10
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9 | elsucg 5490 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | 8, 9 | syl5bb 261 |
. . . . . . . . 9
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11 | 6, 10 | bitrd 257 |
. . . . . . . 8
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12 | 11 | biimpa 487 |
. . . . . . 7
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13 | 12 | ord 379 |
. . . . . 6
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14 | 3, 13 | mpi 20 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 14 | ex 436 |
. . . 4
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16 | 15 | necon3ad 2637 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 2, 16 | mpd 15 |
. 2
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18 | ltrelpi 9314 |
. . . . 5
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19 | 18 | brel 4883 |
. . . 4
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20 | 19 | simpld 461 |
. . 3
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21 | 20 | con3i 141 |
. 2
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22 | 17, 21 | pm2.61i 168 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pr 4639 ax-un 6583 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-sbc 3268 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-tr 4498 df-eprel 4745 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-om 6693 df-1o 7182 df-ni 9297 df-lti 9300 |
This theorem is referenced by: indpi 9332 pinq 9352 archnq 9405 |
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