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Mirrors > Home > MPE Home > Th. List > Mathboxes > sltsolem1 | Structured version Visualization version Unicode version |
Description: Lemma for sltso 31827. The sign expansion relationship totally orders the surreal signs. (Contributed by Scott Fenton, 8-Jun-2011.) |
Ref | Expression |
---|---|
sltsolem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1n0 7575 | . . . . . . . 8 | |
2 | 1 | neii 2796 | . . . . . . 7 |
3 | eqtr2 2642 | . . . . . . 7 | |
4 | 2, 3 | mto 188 | . . . . . 6 |
5 | 1on 7567 | . . . . . . . . 9 | |
6 | 0elon 5778 | . . . . . . . . 9 | |
7 | df-2o 7561 | . . . . . . . . . . 11 | |
8 | df-1o 7560 | . . . . . . . . . . 11 | |
9 | 7, 8 | eqeq12i 2636 | . . . . . . . . . 10 |
10 | suc11 5831 | . . . . . . . . . 10 | |
11 | 9, 10 | syl5bb 272 | . . . . . . . . 9 |
12 | 5, 6, 11 | mp2an 708 | . . . . . . . 8 |
13 | 1, 12 | nemtbir 2889 | . . . . . . 7 |
14 | eqtr2 2642 | . . . . . . . 8 | |
15 | 14 | ancoms 469 | . . . . . . 7 |
16 | 13, 15 | mto 188 | . . . . . 6 |
17 | nsuceq0 5805 | . . . . . . . 8 | |
18 | 7 | eqeq1i 2627 | . . . . . . . 8 |
19 | 17, 18 | nemtbir 2889 | . . . . . . 7 |
20 | eqtr2 2642 | . . . . . . . 8 | |
21 | 20 | ancoms 469 | . . . . . . 7 |
22 | 19, 21 | mto 188 | . . . . . 6 |
23 | 4, 16, 22 | 3pm3.2ni 31594 | . . . . 5 |
24 | vex 3203 | . . . . . 6 | |
25 | 24, 24 | brtp 31639 | . . . . 5 |
26 | 23, 25 | mtbir 313 | . . . 4 |
27 | 26 | a1i 11 | . . 3 |
28 | vex 3203 | . . . . . . 7 | |
29 | 24, 28 | brtp 31639 | . . . . . 6 |
30 | vex 3203 | . . . . . . 7 | |
31 | 28, 30 | brtp 31639 | . . . . . 6 |
32 | eqtr2 2642 | . . . . . . . . . . . . 13 | |
33 | 2, 32 | mto 188 | . . . . . . . . . . . 12 |
34 | 33 | pm2.21i 116 | . . . . . . . . . . 11 |
35 | 34 | ad2ant2rl 785 | . . . . . . . . . 10 |
36 | 35 | expcom 451 | . . . . . . . . 9 |
37 | 34 | ad2ant2rl 785 | . . . . . . . . . 10 |
38 | 37 | expcom 451 | . . . . . . . . 9 |
39 | 3mix2 1231 | . . . . . . . . . . 11 | |
40 | 39 | ad2ant2rl 785 | . . . . . . . . . 10 |
41 | 40 | ex 450 | . . . . . . . . 9 |
42 | 36, 38, 41 | 3jaod 1392 | . . . . . . . 8 |
43 | eqtr2 2642 | . . . . . . . . . . . . 13 | |
44 | 13, 43 | mto 188 | . . . . . . . . . . . 12 |
45 | 44 | pm2.21i 116 | . . . . . . . . . . 11 |
46 | 45 | ad2ant2lr 784 | . . . . . . . . . 10 |
47 | 46 | ex 450 | . . . . . . . . 9 |
48 | 45 | ad2ant2lr 784 | . . . . . . . . . 10 |
49 | 48 | ex 450 | . . . . . . . . 9 |
50 | eqtr2 2642 | . . . . . . . . . . . . 13 | |
51 | 19, 50 | mto 188 | . . . . . . . . . . . 12 |
52 | 51 | pm2.21i 116 | . . . . . . . . . . 11 |
53 | 52 | ad2ant2lr 784 | . . . . . . . . . 10 |
54 | 53 | ex 450 | . . . . . . . . 9 |
55 | 47, 49, 54 | 3jaod 1392 | . . . . . . . 8 |
56 | 45 | ad2ant2lr 784 | . . . . . . . . . 10 |
57 | 56 | ex 450 | . . . . . . . . 9 |
58 | 45 | ad2ant2lr 784 | . . . . . . . . . 10 |
59 | 58 | ex 450 | . . . . . . . . 9 |
60 | 52 | ad2ant2lr 784 | . . . . . . . . . 10 |
61 | 60 | ex 450 | . . . . . . . . 9 |
62 | 57, 59, 61 | 3jaod 1392 | . . . . . . . 8 |
63 | 42, 55, 62 | 3jaoi 1391 | . . . . . . 7 |
64 | 63 | imp 445 | . . . . . 6 |
65 | 29, 31, 64 | syl2anb 496 | . . . . 5 |
66 | 24, 30 | brtp 31639 | . . . . 5 |
67 | 65, 66 | sylibr 224 | . . . 4 |
68 | 67 | a1i 11 | . . 3 |
69 | 24 | eltp 4230 | . . . . 5 |
70 | 28 | eltp 4230 | . . . . 5 |
71 | eqtr3 2643 | . . . . . . . . . 10 | |
72 | 71 | 3mix2d 1237 | . . . . . . . . 9 |
73 | 72 | ex 450 | . . . . . . . 8 |
74 | 3mix2 1231 | . . . . . . . . . 10 | |
75 | 74 | 3mix1d 1236 | . . . . . . . . 9 |
76 | 75 | ex 450 | . . . . . . . 8 |
77 | 3mix1 1230 | . . . . . . . . . 10 | |
78 | 77 | 3mix1d 1236 | . . . . . . . . 9 |
79 | 78 | ex 450 | . . . . . . . 8 |
80 | 73, 76, 79 | 3jaod 1392 | . . . . . . 7 |
81 | 3mix2 1231 | . . . . . . . . . 10 | |
82 | 81 | 3mix3d 1238 | . . . . . . . . 9 |
83 | 82 | expcom 451 | . . . . . . . 8 |
84 | eqtr3 2643 | . . . . . . . . . 10 | |
85 | 84 | 3mix2d 1237 | . . . . . . . . 9 |
86 | 85 | ex 450 | . . . . . . . 8 |
87 | 3mix3 1232 | . . . . . . . . . 10 | |
88 | 87 | 3mix3d 1238 | . . . . . . . . 9 |
89 | 88 | expcom 451 | . . . . . . . 8 |
90 | 83, 86, 89 | 3jaod 1392 | . . . . . . 7 |
91 | 3mix1 1230 | . . . . . . . . . 10 | |
92 | 91 | 3mix3d 1238 | . . . . . . . . 9 |
93 | 92 | expcom 451 | . . . . . . . 8 |
94 | 3mix3 1232 | . . . . . . . . . 10 | |
95 | 94 | 3mix1d 1236 | . . . . . . . . 9 |
96 | 95 | ex 450 | . . . . . . . 8 |
97 | eqtr3 2643 | . . . . . . . . . 10 | |
98 | 97 | 3mix2d 1237 | . . . . . . . . 9 |
99 | 98 | ex 450 | . . . . . . . 8 |
100 | 93, 96, 99 | 3jaod 1392 | . . . . . . 7 |
101 | 80, 90, 100 | 3jaoi 1391 | . . . . . 6 |
102 | 101 | imp 445 | . . . . 5 |
103 | 69, 70, 102 | syl2anb 496 | . . . 4 |
104 | biid 251 | . . . . 5 | |
105 | 28, 24 | brtp 31639 | . . . . 5 |
106 | 29, 104, 105 | 3orbi123i 1252 | . . . 4 |
107 | 103, 106 | sylibr 224 | . . 3 |
108 | 27, 68, 107 | issoi 5066 | . 2 |
109 | df-tp 4182 | . . 3 | |
110 | soeq2 5055 | . . 3 | |
111 | 109, 110 | ax-mp 5 | . 2 |
112 | 108, 111 | mpbi 220 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3o 1036 w3a 1037 wceq 1483 wcel 1990 cun 3572 c0 3915 csn 4177 cpr 4179 ctp 4181 cop 4183 class class class wbr 4653 wor 5034 con0 5723 csuc 5725 c1o 7553 c2o 7554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-suc 5729 df-1o 7560 df-2o 7561 |
This theorem is referenced by: sltso 31827 |
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