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Mirrors > Home > ILE Home > Th. List > ssorduni | Unicode version |
Description: The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
ssorduni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni2 3584 | . . . . 5 | |
2 | ssel 2939 | . . . . . . . . 9 | |
3 | onelss 4124 | . . . . . . . . 9 | |
4 | 2, 3 | syl6 29 | . . . . . . . 8 |
5 | anc2r 311 | . . . . . . . 8 | |
6 | 4, 5 | syl 14 | . . . . . . 7 |
7 | ssuni 3602 | . . . . . . 7 | |
8 | 6, 7 | syl8 65 | . . . . . 6 |
9 | 8 | rexlimdv 2432 | . . . . 5 |
10 | 1, 9 | syl5bi 141 | . . . 4 |
11 | 10 | ralrimiv 2391 | . . 3 |
12 | dftr3 3858 | . . 3 | |
13 | 11, 12 | sylibr 137 | . 2 |
14 | onelon 4121 | . . . . . . 7 | |
15 | 14 | ex 108 | . . . . . 6 |
16 | 2, 15 | syl6 29 | . . . . 5 |
17 | 16 | rexlimdv 2432 | . . . 4 |
18 | 1, 17 | syl5bi 141 | . . 3 |
19 | 18 | ssrdv 2951 | . 2 |
20 | ordon 4212 | . . 3 | |
21 | trssord 4117 | . . . 4 | |
22 | 21 | 3exp 1103 | . . 3 |
23 | 20, 22 | mpii 39 | . 2 |
24 | 13, 19, 23 | sylc 56 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wcel 1393 wral 2306 wrex 2307 wss 2917 cuni 3580 wtr 3854 word 4099 con0 4100 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 df-tr 3855 df-iord 4103 df-on 4105 |
This theorem is referenced by: ssonuni 4214 orduni 4221 tfrlem8 5934 tfrexlem 5948 |
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