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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 3710 | . . . . 5 | |
2 | ssel 3061 | . . . . . . . . 9 | |
3 | onelss 4279 | . . . . . . . . 9 | |
4 | 2, 3 | syl6 33 | . . . . . . . 8 |
5 | anc2r 326 | . . . . . . . 8 | |
6 | 4, 5 | syl 14 | . . . . . . 7 |
7 | ssuni 3728 | . . . . . . 7 | |
8 | 6, 7 | syl8 71 | . . . . . 6 |
9 | 8 | rexlimdv 2525 | . . . . 5 |
10 | 1, 9 | syl5bi 151 | . . . 4 |
11 | 10 | ralrimiv 2481 | . . 3 |
12 | dftr3 4000 | . . 3 | |
13 | 11, 12 | sylibr 133 | . 2 |
14 | onelon 4276 | . . . . . . 7 | |
15 | 14 | ex 114 | . . . . . 6 |
16 | 2, 15 | syl6 33 | . . . . 5 |
17 | 16 | rexlimdv 2525 | . . . 4 |
18 | 1, 17 | syl5bi 151 | . . 3 |
19 | 18 | ssrdv 3073 | . 2 |
20 | ordon 4372 | . . 3 | |
21 | trssord 4272 | . . . 4 | |
22 | 21 | 3exp 1165 | . . 3 |
23 | 20, 22 | mpii 44 | . 2 |
24 | 13, 19, 23 | sylc 62 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 wral 2393 wrex 2394 wss 3041 cuni 3706 wtr 3996 word 4254 con0 4255 |
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-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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
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-in 3047 df-ss 3054 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 |
This theorem is referenced by: ssonuni 4374 orduni 4381 tfrlem8 6183 tfrexlem 6199 |
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