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Mirrors > Home > MPE Home > Th. List > ordelord | Structured version Visualization version Unicode version |
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.) |
Ref | Expression |
---|---|
ordelord |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2689 | . . . . 5 | |
2 | 1 | anbi2d 740 | . . . 4 |
3 | ordeq 5730 | . . . 4 | |
4 | 2, 3 | imbi12d 334 | . . 3 |
5 | simpll 790 | . . . . . . . . 9 | |
6 | 3anrot 1043 | . . . . . . . . . . . 12 | |
7 | 3anass 1042 | . . . . . . . . . . . 12 | |
8 | 6, 7 | bitr3i 266 | . . . . . . . . . . 11 |
9 | ordtr 5737 | . . . . . . . . . . . 12 | |
10 | trel3 4760 | . . . . . . . . . . . 12 | |
11 | 9, 10 | syl 17 | . . . . . . . . . . 11 |
12 | 8, 11 | syl5bir 233 | . . . . . . . . . 10 |
13 | 12 | impl 650 | . . . . . . . . 9 |
14 | trel 4759 | . . . . . . . . . . . . 13 | |
15 | 9, 14 | syl 17 | . . . . . . . . . . . 12 |
16 | 15 | expcomd 454 | . . . . . . . . . . 11 |
17 | 16 | imp31 448 | . . . . . . . . . 10 |
18 | 17 | adantrl 752 | . . . . . . . . 9 |
19 | simplr 792 | . . . . . . . . 9 | |
20 | ordwe 5736 | . . . . . . . . . 10 | |
21 | wetrep 5107 | . . . . . . . . . 10 | |
22 | 20, 21 | sylan 488 | . . . . . . . . 9 |
23 | 5, 13, 18, 19, 22 | syl13anc 1328 | . . . . . . . 8 |
24 | 23 | ex 450 | . . . . . . 7 |
25 | 24 | pm2.43d 53 | . . . . . 6 |
26 | 25 | alrimivv 1856 | . . . . 5 |
27 | dftr2 4754 | . . . . 5 | |
28 | 26, 27 | sylibr 224 | . . . 4 |
29 | trss 4761 | . . . . . . 7 | |
30 | 9, 29 | syl 17 | . . . . . 6 |
31 | wess 5101 | . . . . . 6 | |
32 | 30, 20, 31 | syl6ci 71 | . . . . 5 |
33 | 32 | imp 445 | . . . 4 |
34 | df-ord 5726 | . . . 4 | |
35 | 28, 33, 34 | sylanbrc 698 | . . 3 |
36 | 4, 35 | vtoclg 3266 | . 2 |
37 | 36 | anabsi7 860 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wal 1481 wceq 1483 wcel 1990 wss 3574 wtr 4752 cep 5028 wwe 5072 word 5722 |
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-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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 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 |
This theorem is referenced by: tron 5746 ordelon 5747 ordtr2 5768 ordtr3OLD 5770 ordintdif 5774 ordsuc 7014 ordsucss 7018 ordsucelsuc 7022 ordsucuniel 7024 limsssuc 7050 smores 7449 smo11 7461 smoord 7462 smoword 7463 smogt 7464 smorndom 7465 rdglim2 7528 oesuclem 7605 ordtypelem3 8425 r1val1 8649 rankr1ag 8665 fin23lem24 9144 onsuct0 32440 dford3 37595 ordpss 38655 |
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