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Mirrors > Home > MPE Home > Th. List > ordsucelsuc | Structured version Visualization version Unicode version |
Description: Membership is inherited by successors. Generalization of Exercise 9 of [TakeutiZaring] p. 42. (Contributed by NM, 22-Jun-1998.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
ordsucelsuc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . 3 | |
2 | ordelord 5745 | . . 3 | |
3 | 1, 2 | jca 554 | . 2 |
4 | simpl 473 | . . 3 | |
5 | ordsuc 7014 | . . . 4 | |
6 | ordelord 5745 | . . . . 5 | |
7 | ordsuc 7014 | . . . . 5 | |
8 | 6, 7 | sylibr 224 | . . . 4 |
9 | 5, 8 | sylanb 489 | . . 3 |
10 | 4, 9 | jca 554 | . 2 |
11 | ordsseleq 5752 | . . . . . . . 8 | |
12 | 7, 11 | sylanb 489 | . . . . . . 7 |
13 | 12 | ancoms 469 | . . . . . 6 |
14 | 13 | adantl 482 | . . . . 5 |
15 | ordsucss 7018 | . . . . . . 7 | |
16 | 15 | ad2antrl 764 | . . . . . 6 |
17 | sucssel 5819 | . . . . . . 7 | |
18 | 17 | adantr 481 | . . . . . 6 |
19 | 16, 18 | impbid 202 | . . . . 5 |
20 | sucexb 7009 | . . . . . . 7 | |
21 | elsucg 5792 | . . . . . . 7 | |
22 | 20, 21 | sylbi 207 | . . . . . 6 |
23 | 22 | adantr 481 | . . . . 5 |
24 | 14, 19, 23 | 3bitr4d 300 | . . . 4 |
25 | 24 | ex 450 | . . 3 |
26 | elex 3212 | . . . . 5 | |
27 | elex 3212 | . . . . . 6 | |
28 | 27, 20 | sylibr 224 | . . . . 5 |
29 | 26, 28 | pm5.21ni 367 | . . . 4 |
30 | 29 | a1d 25 | . . 3 |
31 | 25, 30 | pm2.61i 176 | . 2 |
32 | 3, 10, 31 | pm5.21nd 941 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wceq 1483 wcel 1990 cvv 3200 wss 3574 word 5722 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-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-suc 5729 |
This theorem is referenced by: ordsucsssuc 7023 oalimcl 7640 omlimcl 7658 pssnn 8178 cantnflt 8569 cantnfp1lem3 8577 r1pw 8708 r1pwALT 8709 rankelpr 8736 rankelop 8737 rankxplim3 8744 infpssrlem4 9128 axdc3lem2 9273 axdc3lem4 9275 grur1a 9641 bnj570 30975 bnj1001 31028 nosupno 31849 |
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