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Mirrors > Home > ILE Home > Th. List > limom | Unicode version |
Description: Omega is a limit ordinal. Theorem 2.8 of [BellMachover] p. 473. (Contributed by NM, 26-Mar-1995.) (Proof rewritten by Jim Kingdon, 5-Jan-2019.) |
Ref | Expression |
---|---|
limom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordom 4490 | . 2 | |
2 | peano1 4478 | . 2 | |
3 | vex 2663 | . . . . . . . . 9 | |
4 | 3 | sucex 4385 | . . . . . . . 8 |
5 | 4 | isseti 2668 | . . . . . . 7 |
6 | peano2 4479 | . . . . . . . . 9 | |
7 | 3 | sucid 4309 | . . . . . . . . 9 |
8 | 6, 7 | jctil 310 | . . . . . . . 8 |
9 | eleq2 2181 | . . . . . . . . 9 | |
10 | eleq1 2180 | . . . . . . . . 9 | |
11 | 9, 10 | anbi12d 464 | . . . . . . . 8 |
12 | 8, 11 | syl5ibr 155 | . . . . . . 7 |
13 | 5, 12 | eximii 1566 | . . . . . 6 |
14 | 13 | 19.37aiv 1638 | . . . . 5 |
15 | eluni 3709 | . . . . 5 | |
16 | 14, 15 | sylibr 133 | . . . 4 |
17 | 16 | ssriv 3071 | . . 3 |
18 | orduniss 4317 | . . . 4 | |
19 | 1, 18 | ax-mp 5 | . . 3 |
20 | 17, 19 | eqssi 3083 | . 2 |
21 | dflim2 4262 | . 2 | |
22 | 1, 2, 20, 21 | mpbir3an 1148 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wex 1453 wcel 1465 wss 3041 c0 3333 cuni 3706 word 4254 wlim 4256 csuc 4257 com 4474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-tr 3997 df-iord 4258 df-ilim 4261 df-suc 4263 df-iom 4475 |
This theorem is referenced by: freccllem 6267 frecfcllem 6269 frecsuclem 6271 |
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