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Mirrors > Home > MPE Home > Th. List > tfinds2 | Structured version Visualization version 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 4790 | . . . 4 | |
3 | tfinds2.1 | . . . . 5 | |
4 | 3 | imbi2d 330 | . . . 4 |
5 | 2, 4 | sbcie 3470 | . . 3 |
6 | 1, 5 | mpbir 221 | . 2 |
7 | vex 3203 | . . . . . 6 | |
8 | tfinds2.5 | . . . . . . . 8 | |
9 | 8 | a2d 29 | . . . . . . 7 |
10 | 9 | sbcth 3450 | . . . . . 6 |
11 | 7, 10 | ax-mp 5 | . . . . 5 |
12 | sbcimg 3477 | . . . . . 6 | |
13 | 7, 12 | ax-mp 5 | . . . . 5 |
14 | 11, 13 | mpbi 220 | . . . 4 |
15 | sbcel1v 3495 | . . . 4 | |
16 | sbcimg 3477 | . . . . 5 | |
17 | 7, 16 | ax-mp 5 | . . . 4 |
18 | 14, 15, 17 | 3imtr3i 280 | . . 3 |
19 | tfinds2.2 | . . . . . . 7 | |
20 | 19 | bicomd 213 | . . . . . 6 |
21 | 20 | equcoms 1947 | . . . . 5 |
22 | 21 | imbi2d 330 | . . . 4 |
23 | 7, 22 | sbcie 3470 | . . 3 |
24 | vex 3203 | . . . . . . 7 | |
25 | 24 | sucex 7011 | . . . . . 6 |
26 | tfinds2.3 | . . . . . . 7 | |
27 | 26 | imbi2d 330 | . . . . . 6 |
28 | 25, 27 | sbcie 3470 | . . . . 5 |
29 | 28 | sbcbii 3491 | . . . 4 |
30 | suceq 5790 | . . . . 5 | |
31 | 30 | sbcco2 3459 | . . . 4 |
32 | 29, 31 | bitr3i 266 | . . 3 |
33 | 18, 23, 32 | 3imtr3g 284 | . 2 |
34 | sbsbc 3439 | . . . 4 | |
35 | 22 | sbralie 3184 | . . . 4 |
36 | 34, 35 | bitr3i 266 | . . 3 |
37 | r19.21v 2960 | . . . . . . . 8 | |
38 | tfinds2.6 | . . . . . . . . 9 | |
39 | 38 | a2d 29 | . . . . . . . 8 |
40 | 37, 39 | syl5bi 232 | . . . . . . 7 |
41 | 40 | sbcth 3450 | . . . . . 6 |
42 | 24, 41 | ax-mp 5 | . . . . 5 |
43 | sbcimg 3477 | . . . . . 6 | |
44 | 24, 43 | ax-mp 5 | . . . . 5 |
45 | 42, 44 | mpbi 220 | . . . 4 |
46 | limeq 5735 | . . . . 5 | |
47 | 24, 46 | sbcie 3470 | . . . 4 |
48 | sbcimg 3477 | . . . . 5 | |
49 | 24, 48 | ax-mp 5 | . . . 4 |
50 | 45, 47, 49 | 3imtr3i 280 | . . 3 |
51 | 36, 50 | syl5bir 233 | . 2 |
52 | 6, 33, 51 | tfindes 7062 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wceq 1483 wsb 1880 wcel 1990 wral 2912 cvv 3200 wsbc 3435 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: inar1 9597 grur1a 9641 |
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