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Mirrors > Home > ILE Home > Th. List > ordiso | Unicode version |
Description: Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
ordiso |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resiexg 4653 | . . . . 5 | |
2 | isoid 5450 | . . . . 5 | |
3 | isoeq1 5441 | . . . . . 6 | |
4 | 3 | spcegv 2641 | . . . . 5 |
5 | 1, 2, 4 | mpisyl 1335 | . . . 4 |
6 | 5 | adantr 261 | . . 3 |
7 | isoeq5 5445 | . . . 4 | |
8 | 7 | exbidv 1706 | . . 3 |
9 | 6, 8 | syl5ibcom 144 | . 2 |
10 | eloni 4112 | . . . 4 | |
11 | eloni 4112 | . . . 4 | |
12 | ordiso2 6357 | . . . . . 6 | |
13 | 12 | 3coml 1111 | . . . . 5 |
14 | 13 | 3expia 1106 | . . . 4 |
15 | 10, 11, 14 | syl2an 273 | . . 3 |
16 | 15 | exlimdv 1700 | . 2 |
17 | 9, 16 | impbid 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cvv 2557 cep 4024 cid 4025 word 4099 con0 4100 cres 4347 wiso 4903 |
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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-iord 4103 df-on 4105 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-isom 4911 |
This theorem is referenced by: (None) |
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