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Mirrors > Home > NFE Home > Th. List > lenltfin | Unicode version |
Description: Less than or equal is the same as negated less than. (Contributed by SF, 2-Feb-2015.) |
Ref | Expression |
---|---|
lenltfin | Nn Nn fin fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltfinirr 4457 | . . . . . 6 Nn fin | |
2 | 1 | adantr 451 | . . . . 5 Nn Nn fin |
3 | 2 | adantr 451 | . . . 4 Nn Nn fin fin |
4 | leltfintr 4458 | . . . . . 6 Nn Nn Nn fin fin fin | |
5 | 4 | 3anidm13 1240 | . . . . 5 Nn Nn fin fin fin |
6 | 5 | expdimp 426 | . . . 4 Nn Nn fin fin fin |
7 | 3, 6 | mtod 168 | . . 3 Nn Nn fin fin |
8 | 7 | ex 423 | . 2 Nn Nn fin fin |
9 | nulge 4456 | . . . . . 6 Nn Nn fin | |
10 | 9 | ancoms 439 | . . . . 5 Nn Nn fin |
11 | eleq1 2413 | . . . . . . 7 Nn Nn | |
12 | 11 | anbi2d 684 | . . . . . 6 Nn Nn Nn Nn |
13 | opkeq2 4060 | . . . . . . 7 | |
14 | 13 | eleq1d 2419 | . . . . . 6 fin fin |
15 | 12, 14 | imbi12d 311 | . . . . 5 Nn Nn fin Nn Nn fin |
16 | 10, 15 | mpbiri 224 | . . . 4 Nn Nn fin |
17 | 16 | a1dd 42 | . . 3 Nn Nn fin fin |
18 | simplr 731 | . . . . . . . 8 Nn Nn Nn | |
19 | simpll 730 | . . . . . . . 8 Nn Nn Nn | |
20 | simpr 447 | . . . . . . . 8 Nn Nn | |
21 | ltfintri 4466 | . . . . . . . 8 Nn Nn fin fin | |
22 | 18, 19, 20, 21 | syl3anc 1182 | . . . . . . 7 Nn Nn fin fin |
23 | 3orass 937 | . . . . . . 7 fin fin fin fin | |
24 | 22, 23 | sylib 188 | . . . . . 6 Nn Nn fin fin |
25 | 24 | ord 366 | . . . . 5 Nn Nn fin fin |
26 | lefinrflx 4467 | . . . . . . . . 9 Nn fin | |
27 | 26 | adantr 451 | . . . . . . . 8 Nn Nn fin |
28 | opkeq2 4060 | . . . . . . . . 9 | |
29 | 28 | eleq1d 2419 | . . . . . . . 8 fin fin |
30 | 27, 29 | syl5ibrcom 213 | . . . . . . 7 Nn Nn fin |
31 | 30 | adantr 451 | . . . . . 6 Nn Nn fin |
32 | ltlefin 4468 | . . . . . . 7 Nn Nn fin fin | |
33 | 32 | adantr 451 | . . . . . 6 Nn Nn fin fin |
34 | 31, 33 | jaod 369 | . . . . 5 Nn Nn fin fin |
35 | 25, 34 | syld 40 | . . . 4 Nn Nn fin fin |
36 | 35 | expcom 424 | . . 3 Nn Nn fin fin |
37 | 17, 36 | pm2.61ine 2592 | . 2 Nn Nn fin fin |
38 | 8, 37 | impbid 183 | 1 Nn Nn fin fin |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wo 357 wa 358 w3o 933 wceq 1642 wcel 1710 wne 2516 c0 3550 copk 4057 Nn cnnc 4373 fin clefin 4432 fin cltfin 4433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-0c 4377 df-addc 4378 df-nnc 4379 df-lefin 4440 df-ltfin 4441 |
This theorem is referenced by: tfinlefin 4502 vfinspsslem1 4550 |
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