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Mirrors > Home > MPE Home > Th. List > tfindsg | Structured version Visualization version Unicode version |
Description: Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. The basis of this version is an arbitrary ordinal instead of zero. Remark in [TakeutiZaring] p. 57. (Contributed by NM, 5-Mar-2004.) |
Ref | Expression |
---|---|
tfindsg.1 | |
tfindsg.2 | |
tfindsg.3 | |
tfindsg.4 | |
tfindsg.5 | |
tfindsg.6 | |
tfindsg.7 |
Ref | Expression |
---|---|
tfindsg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 3627 | . . . . . . 7 | |
2 | 1 | adantl 482 | . . . . . 6 |
3 | eqeq2 2633 | . . . . . . . 8 | |
4 | tfindsg.1 | . . . . . . . 8 | |
5 | 3, 4 | syl6bir 244 | . . . . . . 7 |
6 | 5 | imp 445 | . . . . . 6 |
7 | 2, 6 | imbi12d 334 | . . . . 5 |
8 | 1 | imbi1d 331 | . . . . . 6 |
9 | ss0 3974 | . . . . . . . . 9 | |
10 | 9 | con3i 150 | . . . . . . . 8 |
11 | 10 | pm2.21d 118 | . . . . . . 7 |
12 | 11 | pm5.74d 262 | . . . . . 6 |
13 | 8, 12 | sylan9bbr 737 | . . . . 5 |
14 | 7, 13 | pm2.61ian 831 | . . . 4 |
15 | 14 | imbi2d 330 | . . 3 |
16 | sseq2 3627 | . . . . 5 | |
17 | tfindsg.2 | . . . . 5 | |
18 | 16, 17 | imbi12d 334 | . . . 4 |
19 | 18 | imbi2d 330 | . . 3 |
20 | sseq2 3627 | . . . . 5 | |
21 | tfindsg.3 | . . . . 5 | |
22 | 20, 21 | imbi12d 334 | . . . 4 |
23 | 22 | imbi2d 330 | . . 3 |
24 | sseq2 3627 | . . . . 5 | |
25 | tfindsg.4 | . . . . 5 | |
26 | 24, 25 | imbi12d 334 | . . . 4 |
27 | 26 | imbi2d 330 | . . 3 |
28 | tfindsg.5 | . . . 4 | |
29 | 28 | a1d 25 | . . 3 |
30 | vex 3203 | . . . . . . . . . . . . . 14 | |
31 | 30 | sucex 7011 | . . . . . . . . . . . . 13 |
32 | 31 | eqvinc 3330 | . . . . . . . . . . . 12 |
33 | 28, 4 | syl5ibr 236 | . . . . . . . . . . . . . 14 |
34 | 21 | biimpd 219 | . . . . . . . . . . . . . 14 |
35 | 33, 34 | sylan9r 690 | . . . . . . . . . . . . 13 |
36 | 35 | exlimiv 1858 | . . . . . . . . . . . 12 |
37 | 32, 36 | sylbi 207 | . . . . . . . . . . 11 |
38 | 37 | eqcoms 2630 | . . . . . . . . . 10 |
39 | 38 | imim2i 16 | . . . . . . . . 9 |
40 | 39 | a1d 25 | . . . . . . . 8 |
41 | 40 | com4r 94 | . . . . . . 7 |
42 | 41 | adantl 482 | . . . . . 6 |
43 | df-ne 2795 | . . . . . . . . 9 | |
44 | 43 | anbi2i 730 | . . . . . . . 8 |
45 | annim 441 | . . . . . . . 8 | |
46 | 44, 45 | bitri 264 | . . . . . . 7 |
47 | onsssuc 5813 | . . . . . . . . . 10 | |
48 | suceloni 7013 | . . . . . . . . . . 11 | |
49 | onelpss 5764 | . . . . . . . . . . 11 | |
50 | 48, 49 | sylan2 491 | . . . . . . . . . 10 |
51 | 47, 50 | bitrd 268 | . . . . . . . . 9 |
52 | 51 | ancoms 469 | . . . . . . . 8 |
53 | tfindsg.6 | . . . . . . . . . . . 12 | |
54 | 53 | ex 450 | . . . . . . . . . . 11 |
55 | ax-1 6 | . . . . . . . . . . 11 | |
56 | 54, 55 | syl8 76 | . . . . . . . . . 10 |
57 | 56 | a2d 29 | . . . . . . . . 9 |
58 | 57 | com23 86 | . . . . . . . 8 |
59 | 52, 58 | sylbird 250 | . . . . . . 7 |
60 | 46, 59 | syl5bir 233 | . . . . . 6 |
61 | 42, 60 | pm2.61d 170 | . . . . 5 |
62 | 61 | ex 450 | . . . 4 |
63 | 62 | a2d 29 | . . 3 |
64 | pm2.27 42 | . . . . . . . . 9 | |
65 | 64 | ralimdv 2963 | . . . . . . . 8 |
66 | 65 | ad2antlr 763 | . . . . . . 7 |
67 | tfindsg.7 | . . . . . . 7 | |
68 | 66, 67 | syld 47 | . . . . . 6 |
69 | 68 | exp31 630 | . . . . 5 |
70 | 69 | com3l 89 | . . . 4 |
71 | 70 | com4t 93 | . . 3 |
72 | 15, 19, 23, 27, 29, 63, 71 | tfinds 7059 | . 2 |
73 | 72 | imp31 448 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 wne 2794 wral 2912 wss 3574 c0 3915 con0 5723 wlim 5724 csuc 5725 |
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-lim 5728 df-suc 5729 |
This theorem is referenced by: tfindsg2 7061 oaordi 7626 infensuc 8138 r1ordg 8641 |
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