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Mirrors > Home > ILE Home > Th. List > onsucuni2 | Unicode version |
Description: A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
onsucuni2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2220 | . . . . . 6 | |
2 | 1 | biimpac 296 | . . . . 5 |
3 | sucelon 4462 | . . . . . . 7 | |
4 | eloni 4335 | . . . . . . . . . 10 | |
5 | ordtr 4338 | . . . . . . . . . 10 | |
6 | 4, 5 | syl 14 | . . . . . . . . 9 |
7 | unisucg 4374 | . . . . . . . . 9 | |
8 | 6, 7 | mpbid 146 | . . . . . . . 8 |
9 | suceq 4362 | . . . . . . . 8 | |
10 | 8, 9 | syl 14 | . . . . . . 7 |
11 | 3, 10 | sylbir 134 | . . . . . 6 |
12 | eloni 4335 | . . . . . . . 8 | |
13 | ordtr 4338 | . . . . . . . 8 | |
14 | 12, 13 | syl 14 | . . . . . . 7 |
15 | unisucg 4374 | . . . . . . 7 | |
16 | 14, 15 | mpbid 146 | . . . . . 6 |
17 | 11, 16 | eqtr4d 2193 | . . . . 5 |
18 | 2, 17 | syl 14 | . . . 4 |
19 | unieq 3781 | . . . . . 6 | |
20 | suceq 4362 | . . . . . 6 | |
21 | 19, 20 | syl 14 | . . . . 5 |
22 | suceq 4362 | . . . . . 6 | |
23 | 22 | unieqd 3783 | . . . . 5 |
24 | 21, 23 | eqeq12d 2172 | . . . 4 |
25 | 18, 24 | syl5ibr 155 | . . 3 |
26 | 25 | anabsi7 571 | . 2 |
27 | eloni 4335 | . . . . 5 | |
28 | ordtr 4338 | . . . . 5 | |
29 | 27, 28 | syl 14 | . . . 4 |
30 | unisucg 4374 | . . . 4 | |
31 | 29, 30 | mpbid 146 | . . 3 |
32 | 31 | adantr 274 | . 2 |
33 | 26, 32 | eqtrd 2190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 cuni 3772 wtr 4062 word 4322 con0 4323 csuc 4325 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3773 df-tr 4063 df-iord 4326 df-on 4328 df-suc 4331 |
This theorem is referenced by: nnsucpred 4576 nnpredcl 4582 |
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