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| Description: The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values. |
| Ref | Expression |
|---|---|
| rdglim2 |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rdglim 5156 |
. 2
| |
| 2 | limord 3723 |
. . . . . . . . . . 11
| |
| 3 | ordelord 3680 |
. . . . . . . . . . . . 13
| |
| 4 | 3 | ex 402 |
. . . . . . . . . . . 12
|
| 5 | visset 2295 |
. . . . . . . . . . . . 13
| |
| 6 | 5 | elon 3666 |
. . . . . . . . . . . 12
|
| 7 | 4, 6 | syl6ibr 230 |
. . . . . . . . . . 11
|
| 8 | 2, 7 | syl 12 |
. . . . . . . . . 10
|
| 9 | rdgfnon 5147 |
. . . . . . . . . . . 12
| |
| 10 | visset 2295 |
. . . . . . . . . . . . 13
| |
| 11 | 10 | fnopfvb 4713 |
. . . . . . . . . . . 12
|
| 12 | 9, 11 | mpan 759 |
. . . . . . . . . . 11
|
| 13 | eqcom 1886 |
. . . . . . . . . . 11
| |
| 14 | 12, 13 | syl5bb 591 |
. . . . . . . . . 10
|
| 15 | 8, 14 | syl6 25 |
. . . . . . . . 9
|
| 16 | 15 | pm5.32d 709 |
. . . . . . . 8
|
| 17 | 16 | exbidv 1657 |
. . . . . . 7
|
| 18 | df-rex 2110 |
. . . . . . 7
| |
| 19 | 17, 18 | syl5rbb 592 |
. . . . . 6
|
| 20 | 19 | abbidv 2008 |
. . . . 5
|
| 21 | dfima3 4267 |
. . . . 5
| |
| 22 | 20, 21 | syl5eq 1940 |
. . . 4
|
| 23 | 22 | unieqd 3188 |
. . 3
|
| 24 | 23 | adantl 424 |
. 2
|
| 25 | 1, 24 | eqtrd 1925 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: rdglim2a 5158 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 1304 ax-gen 1305 ax-8 1306 ax-9 1307 ax-10 1308 ax-11 1309 ax-12 1310 ax-13 1311 ax-14 1312 ax-17 1317 ax-4 1319 ax-5o 1321 ax-6o 1324 ax-9o 1481 ax-10o 1500 ax-16 1580 ax-11o 1588 ax-ext 1865 ax-rep 3428 ax-sep 3438 ax-nul 3445 ax-pow 3481 ax-pr 3524 ax-un 3790 |
| This theorem depends on definitions: df-bi 164 df-or 241 df-an 242 df-3or 859 df-3an 860 df-ex 1327 df-sb 1536 df-eu 1775 df-mo 1776 df-clab 1872 df-cleq 1877 df-clel 1880 df-ne 2019 df-ral 2109 df-rex 2110 df-rab 2112 df-v 2294 df-sbc 2454 df-dif 2597 df-un 2600 df-in 2603 df-ss 2605 df-pss 2607 df-nul 2876 df-if 2983 df-pw 3035 df-sn 3049 df-pr 3050 df-tp 3052 df-op 3053 df-uni 3178 df-iun 3257 df-br 3339 df-opab 3396 df-tr 3412 df-eprel 3583 df-id 3586 df-po 3591 df-so 3604 df-fr 3625 df-we 3644 df-ord 3660 df-on 3661 df-lim 3662 df-suc 3663 df-xp 4000 df-rel 4001 df-cnv 4002 df-co 4003 df-dm 4004 df-rn 4005 df-res 4006 df-ima 4007 df-fun 4008 df-fn 4009 df-fv 4014 df-rdg 5140 |