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Mirrors > Home > MPE Home > Th. List > ordom | Structured version Visualization version Unicode version |
Description: Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ordom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr2 4754 | . . 3 | |
2 | onelon 5748 | . . . . . . . 8 | |
3 | 2 | expcom 451 | . . . . . . 7 |
4 | limord 5784 | . . . . . . . . . . . 12 | |
5 | ordtr 5737 | . . . . . . . . . . . 12 | |
6 | trel 4759 | . . . . . . . . . . . 12 | |
7 | 4, 5, 6 | 3syl 18 | . . . . . . . . . . 11 |
8 | 7 | expd 452 | . . . . . . . . . 10 |
9 | 8 | com12 32 | . . . . . . . . 9 |
10 | 9 | a2d 29 | . . . . . . . 8 |
11 | 10 | alimdv 1845 | . . . . . . 7 |
12 | 3, 11 | anim12d 586 | . . . . . 6 |
13 | elom 7068 | . . . . . 6 | |
14 | elom 7068 | . . . . . 6 | |
15 | 12, 13, 14 | 3imtr4g 285 | . . . . 5 |
16 | 15 | imp 445 | . . . 4 |
17 | 16 | ax-gen 1722 | . . 3 |
18 | 1, 17 | mpgbir 1726 | . 2 |
19 | omsson 7069 | . 2 | |
20 | ordon 6982 | . 2 | |
21 | trssord 5740 | . 2 | |
22 | 18, 19, 20, 21 | mp3an 1424 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wal 1481 wcel 1990 wss 3574 wtr 4752 word 5722 con0 5723 wlim 5724 com 7065 |
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-lim 5728 df-suc 5729 df-om 7066 |
This theorem is referenced by: elnn 7075 omon 7076 limom 7080 ssnlim 7083 omsinds 7084 peano5 7089 nnarcl 7696 nnawordex 7717 oaabslem 7723 oaabs2 7725 omabslem 7726 onomeneq 8150 ominf 8172 findcard3 8203 nnsdomg 8219 dffi3 8337 wofib 8450 alephgeom 8905 iscard3 8916 iunfictbso 8937 unctb 9027 ackbij2lem1 9041 ackbij1lem3 9044 ackbij1lem18 9059 ackbij2 9065 cflim2 9085 fin23lem26 9147 fin23lem23 9148 fin23lem27 9150 fin67 9217 alephexp1 9401 pwfseqlem3 9482 pwcdandom 9489 winainflem 9515 wunex2 9560 om2uzoi 12754 ltweuz 12760 fz1isolem 13245 mreexexdOLD 16309 1stcrestlem 21255 hfuni 32291 hfninf 32293 finxpreclem4 33231 |
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