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Mirrors > Home > MPE Home > Th. List > tfinds2 | Unicode version |
Description: Transfinite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.) |
Ref | Expression |
---|---|
tfinds2.1 | |
tfinds2.2 | |
tfinds2.3 | |
tfinds2.4 | |
tfinds2.5 | |
tfinds2.6 |
Ref | Expression |
---|---|
tfinds2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfinds2.4 | . . 3 | |
2 | 0ex 4582 | . . . 4 | |
3 | tfinds2.1 | . . . . 5 | |
4 | 3 | imbi2d 316 | . . . 4 |
5 | 2, 4 | sbcie 3362 | . . 3 |
6 | 1, 5 | mpbir 209 | . 2 |
7 | vex 3112 | . . . . . 6 | |
8 | tfinds2.5 | . . . . . . . 8 | |
9 | 8 | a2d 26 | . . . . . . 7 |
10 | 9 | sbcth 3342 | . . . . . 6 |
11 | 7, 10 | ax-mp 5 | . . . . 5 |
12 | sbcimg 3369 | . . . . . 6 | |
13 | 7, 12 | ax-mp 5 | . . . . 5 |
14 | 11, 13 | mpbi 208 | . . . 4 |
15 | sbcel1v 3392 | . . . 4 | |
16 | sbcimg 3369 | . . . . 5 | |
17 | 7, 16 | ax-mp 5 | . . . 4 |
18 | 14, 15, 17 | 3imtr3i 265 | . . 3 |
19 | tfinds2.2 | . . . . . . 7 | |
20 | 19 | bicomd 201 | . . . . . 6 |
21 | 20 | equcoms 1795 | . . . . 5 |
22 | 21 | imbi2d 316 | . . . 4 |
23 | 7, 22 | sbcie 3362 | . . 3 |
24 | vex 3112 | . . . . . . 7 | |
25 | 24 | sucex 6646 | . . . . . 6 |
26 | tfinds2.3 | . . . . . . 7 | |
27 | 26 | imbi2d 316 | . . . . . 6 |
28 | 25, 27 | sbcie 3362 | . . . . 5 |
29 | 28 | sbcbii 3387 | . . . 4 |
30 | suceq 4948 | . . . . 5 | |
31 | 30 | sbcco2 3351 | . . . 4 |
32 | 29, 31 | bitr3i 251 | . . 3 |
33 | 18, 23, 32 | 3imtr3g 269 | . 2 |
34 | sbsbc 3331 | . . . 4 | |
35 | 22 | sbralie 3097 | . . . 4 |
36 | 34, 35 | bitr3i 251 | . . 3 |
37 | r19.21v 2862 | . . . . . . . 8 | |
38 | tfinds2.6 | . . . . . . . . 9 | |
39 | 38 | a2d 26 | . . . . . . . 8 |
40 | 37, 39 | syl5bi 217 | . . . . . . 7 |
41 | 40 | sbcth 3342 | . . . . . 6 |
42 | 24, 41 | ax-mp 5 | . . . . 5 |
43 | sbcimg 3369 | . . . . . 6 | |
44 | 24, 43 | ax-mp 5 | . . . . 5 |
45 | 42, 44 | mpbi 208 | . . . 4 |
46 | limeq 4895 | . . . . 5 | |
47 | 24, 46 | sbcie 3362 | . . . 4 |
48 | sbcimg 3369 | . . . . 5 | |
49 | 24, 48 | ax-mp 5 | . . . 4 |
50 | 45, 47, 49 | 3imtr3i 265 | . . 3 |
51 | 36, 50 | syl5bir 218 | . 2 |
52 | 6, 33, 51 | tfindes 6697 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 <-> wb 184
= wceq 1395 [ wsb 1739 e. wcel 1818
A. wral 2807 cvv 3109
[. wsbc 3327 c0 3784 con0 4883 Lim wlim 4884 suc csuc 4885 |
This theorem is referenced by: inar1 9174 grur1a 9218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pr 4691 ax-un 6592 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-sbc 3328 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-br 4453 df-opab 4511 df-tr 4546 df-eprel 4796 df-po 4805 df-so 4806 df-fr 4843 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 |
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