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Related theorems Unicode version |
| Description: Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42. |
| Ref | Expression |
|---|---|
| ordsucelsuc |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ordsseleq 3033 |
. . . . . . . . . . . 12
| |
| 2 | ordsuc 3122 |
. . . . . . . . . . . 12
| |
| 3 | 1, 2 | sylanb 460 |
. . . . . . . . . . 11
|
| 4 | 3 | adantl 397 |
. . . . . . . . . 10
|
| 5 | ordsucss 3126 |
. . . . . . . . . . . 12
| |
| 6 | 5 | ad2antll 416 |
. . . . . . . . . . 11
|
| 7 | sucssel 3127 |
. . . . . . . . . . . 12
| |
| 8 | 7 | adantr 398 |
. . . . . . . . . . 11
|
| 9 | 6, 8 | impbid 527 |
. . . . . . . . . 10
|
| 10 | sucexb 3105 |
. . . . . . . . . . . 12
| |
| 11 | elsucg 3093 |
. . . . . . . . . . . 12
| |
| 12 | 10, 11 | sylbi 206 |
. . . . . . . . . . 11
|
| 13 | 12 | adantr 398 |
. . . . . . . . . 10
|
| 14 | 4, 9, 13 | 3bitr4d 561 |
. . . . . . . . 9
|
| 15 | 14 | ex 380 |
. . . . . . . 8
|
| 16 | elisset 1864 |
. . . . . . . . . 10
| |
| 17 | elisset 1864 |
. . . . . . . . . . 11
| |
| 18 | 17, 10 | sylibr 207 |
. . . . . . . . . 10
|
| 19 | 16, 18 | pm5.21ni 690 |
. . . . . . . . 9
|
| 20 | 19 | a1d 12 |
. . . . . . . 8
|
| 21 | 15, 20 | pm2.61i 132 |
. . . . . . 7
|
| 22 | 21 | biimpd 160 |
. . . . . 6
|
| 23 | ordelord 3027 |
. . . . . 6
| |
| 24 | 22, 23 | sylan 459 |
. . . . 5
|
| 25 | 24 | exp31 385 |
. . . 4
|
| 26 | 25 | pm2.43a 68 |
. . 3
|
| 27 | 26 | pm2.43d 66 |
. 2
|
| 28 | 21 | biimprd 161 |
. . . . . 6
|
| 29 | ordelord 3027 |
. . . . . . . 8
| |
| 30 | 29, 2 | sylibr 207 |
. . . . . . 7
|
| 31 | ordsuc 3122 |
. . . . . . 7
| |
| 32 | 30, 31 | sylanb 460 |
. . . . . 6
|
| 33 | 28, 32 | sylan 459 |
. . . . 5
|
| 34 | 33 | exp31 385 |
. . . 4
|
| 35 | 34 | pm2.43a 68 |
. . 3
|
| 36 | 35 | pm2.43d 66 |
. 2
|
| 37 | 27, 36 | impbid 527 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: ordsucsssuc 3131 oalimcl 4252 omlimcl 4267 pssnn 4599 r1pw 4748 rankelpr 4770 rankelop 4771 rankxplim3 4776 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 1003 ax-gen 1004 ax-8 1005 ax-10 1007 ax-11 1008 ax-12 1009 ax-13 1010 ax-14 1011 ax-17 1012 ax-4 1014 ax-5o 1016 ax-6o 1019 ax-9o 1164 ax-10o 1182 ax-16 1252 ax-11o 1260 ax-ext 1504 ax-sep 2758 ax-pow 2798 ax-pr 2835 ax-un 2922 |
| This theorem depends on definitions: df-bi 154 df-or 231 df-an 232 df-3or 788 df-3an 789 df-ex 1022 df-sb 1214 df-eu 1424 df-mo 1425 df-clab 1510 df-cleq 1515 df-clel 1518 df-ne 1634 df-ral 1696 df-rex 1697 df-v 1859 df-dif 2100 df-un 2101 df-in 2102 df-ss 2104 df-nul 2332 df-pw 2454 df-sn 2464 df-pr 2465 df-tp 2467 df-op 2468 df-uni 2558 df-br 2675 df-opab 2722 df-tr 2736 df-eprel 2888 df-po 2896 df-so 2906 df-fr 2974 df-we 2991 df-ord 3008 df-on 3009 df-suc 3011 |