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Mirrors > Home > MPE Home > Th. List > oeordsuc | Structured version Visualization version Unicode version |
Description: Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) |
Ref | Expression |
---|---|
oeordsuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onelon 5748 | . . . 4 | |
2 | 1 | ex 450 | . . 3 |
3 | 2 | adantr 481 | . 2 |
4 | oewordri 7672 | . . . . . . . . . . 11 | |
5 | 4 | 3adant1 1079 | . . . . . . . . . 10 |
6 | oecl 7617 | . . . . . . . . . . . 12 | |
7 | 6 | 3adant2 1080 | . . . . . . . . . . 11 |
8 | oecl 7617 | . . . . . . . . . . . 12 | |
9 | 8 | 3adant1 1079 | . . . . . . . . . . 11 |
10 | simp1 1061 | . . . . . . . . . . 11 | |
11 | omwordri 7652 | . . . . . . . . . . 11 | |
12 | 7, 9, 10, 11 | syl3anc 1326 | . . . . . . . . . 10 |
13 | 5, 12 | syld 47 | . . . . . . . . 9 |
14 | oesuc 7607 | . . . . . . . . . . 11 | |
15 | 14 | 3adant2 1080 | . . . . . . . . . 10 |
16 | 15 | sseq1d 3632 | . . . . . . . . 9 |
17 | 13, 16 | sylibrd 249 | . . . . . . . 8 |
18 | ne0i 3921 | . . . . . . . . . . . . . 14 | |
19 | on0eln0 5780 | . . . . . . . . . . . . . 14 | |
20 | 18, 19 | syl5ibr 236 | . . . . . . . . . . . . 13 |
21 | 20 | adantr 481 | . . . . . . . . . . . 12 |
22 | oen0 7666 | . . . . . . . . . . . . 13 | |
23 | 22 | ex 450 | . . . . . . . . . . . 12 |
24 | 21, 23 | syld 47 | . . . . . . . . . . 11 |
25 | simpl 473 | . . . . . . . . . . . . . . 15 | |
26 | 25, 8 | jca 554 | . . . . . . . . . . . . . 14 |
27 | omordi 7646 | . . . . . . . . . . . . . 14 | |
28 | 26, 27 | sylan 488 | . . . . . . . . . . . . 13 |
29 | 28 | ex 450 | . . . . . . . . . . . 12 |
30 | 29 | com23 86 | . . . . . . . . . . 11 |
31 | 24, 30 | mpdd 43 | . . . . . . . . . 10 |
32 | 31 | 3adant1 1079 | . . . . . . . . 9 |
33 | oesuc 7607 | . . . . . . . . . . 11 | |
34 | 33 | 3adant1 1079 | . . . . . . . . . 10 |
35 | 34 | eleq2d 2687 | . . . . . . . . 9 |
36 | 32, 35 | sylibrd 249 | . . . . . . . 8 |
37 | 17, 36 | jcad 555 | . . . . . . 7 |
38 | 37 | 3expa 1265 | . . . . . 6 |
39 | sucelon 7017 | . . . . . . 7 | |
40 | oecl 7617 | . . . . . . . . 9 | |
41 | oecl 7617 | . . . . . . . . 9 | |
42 | ontr2 5772 | . . . . . . . . 9 | |
43 | 40, 41, 42 | syl2an 494 | . . . . . . . 8 |
44 | 43 | anandirs 874 | . . . . . . 7 |
45 | 39, 44 | sylan2b 492 | . . . . . 6 |
46 | 38, 45 | syld 47 | . . . . 5 |
47 | 46 | exp31 630 | . . . 4 |
48 | 47 | com4l 92 | . . 3 |
49 | 48 | imp 445 | . 2 |
50 | 3, 49 | mpdd 43 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wss 3574 c0 3915 con0 5723 csuc 5725 (class class class)co 6650 comu 7558 coe 7559 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 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-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-oexp 7566 |
This theorem is referenced by: (None) |
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