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| Description: Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. |
| Ref | Expression |
|---|---|
| ordom |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dftr2 2672 |
. . 3
| |
| 2 | ordelord 2960 |
. . . . . . . 8
| |
| 3 | nnord 3130 |
. . . . . . . 8
| |
| 4 | 2, 3 | sylan 448 |
. . . . . . 7
|
| 5 | 4 | ancoms 436 |
. . . . . 6
|
| 6 | trel 2677 |
. . . . . . . . . . . . 13
| |
| 7 | 6 | exp3a 375 |
. . . . . . . . . . . 12
|
| 8 | 7 | com12 11 |
. . . . . . . . . . 11
|
| 9 | limord 3018 |
. . . . . . . . . . . 12
| |
| 10 | ordtr 2952 |
. . . . . . . . . . . 12
| |
| 11 | 9, 10 | syl 10 |
. . . . . . . . . . 11
|
| 12 | 8, 11 | syl5 21 |
. . . . . . . . . 10
|
| 13 | 12 | a2d 13 |
. . . . . . . . 9
|
| 14 | 13 | 19.20dv 1284 |
. . . . . . . 8
|
| 15 | visset 1804 |
. . . . . . . . . 10
| |
| 16 | 15 | elom 3124 |
. . . . . . . . 9
|
| 17 | 16 | pm3.27bi 326 |
. . . . . . . 8
|
| 18 | 14, 17 | syl5 21 |
. . . . . . 7
|
| 19 | 18 | imp 350 |
. . . . . 6
|
| 20 | 5, 19 | jca 288 |
. . . . 5
|
| 21 | visset 1804 |
. . . . . 6
| |
| 22 | 21 | elom 3124 |
. . . . 5
|
| 23 | 20, 22 | sylibr 200 |
. . . 4
|
| 24 | 23 | ax-gen 960 |
. . 3
|
| 25 | 1, 24 | mpgbir 985 |
. 2
|
| 26 | omsson 3126 |
. 2
| |
| 27 | ordon 2977 |
. 2
| |
| 28 | trssord 2955 |
. 2
| |
| 29 | 25, 26, 27, 28 | mp3an 913 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: elnn 3132 omon 3133 limom 3136 peano5 3143 ssnlim 3157 nnarcl 4216 oaabslem 4235 oaabs 4236 onomeneq 4498 ominf 4508 omsdomnn 4509 alephgeom 4854 iscard3 4860 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 959 ax-gen 960 ax-8 961 ax-10 963 ax-11 964 ax-12 965 ax-13 966 ax-14 967 ax-17 968 ax-4 970 ax-5o 972 ax-6o 975 ax-9o 1119 ax-10o 1136 ax-16 1206 ax-11o 1213 ax-ext 1452 ax-sep 2693 ax-pow 2732 ax-pr 2769 ax-un 2857 |
| This theorem depends on definitions: df-bi 147 df-or 224 df-an 225 df-3or 774 df-3an 775 df-ex 978 df-sb 1168 df-eu 1375 df-mo 1376 df-clab 1457 df-cleq 1462 df-clel 1465 df-ne 1579 df-ral 1641 df-rex 1642 df-v 1803 df-dif 2039 df-un 2040 df-in 2041 df-ss 2043 df-nul 2271 df-pw 2392 df-sn 2402 df-pr 2403 df-tp 2405 df-op 2406 df-uni 2494 df-br 2610 df-opab 2657 df-tr 2671 df-eprel 2821 df-po 2831 df-so 2841 df-fr 2907 df-we 2924 df-ord 2941 df-on 2942 df-lim 2943 df-om 3122 |