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Mirrors > Home > ILE Home > Th. List > oacl | Unicode version |
Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim Kingdon, 26-Jul-2019.) |
Ref | Expression |
---|---|
oacl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oav 6413 | . 2 | |
2 | id 19 | . . 3 | |
3 | vex 2724 | . . . . . . . 8 | |
4 | suceq 4374 | . . . . . . . . 9 | |
5 | eqid 2164 | . . . . . . . . 9 | |
6 | 3 | sucex 4470 | . . . . . . . . 9 |
7 | 4, 5, 6 | fvmpt 5557 | . . . . . . . 8 |
8 | 3, 7 | ax-mp 5 | . . . . . . 7 |
9 | 8 | eleq1i 2230 | . . . . . 6 |
10 | 9 | ralbii 2470 | . . . . 5 |
11 | suceloni 4472 | . . . . 5 | |
12 | 10, 11 | mprgbir 2522 | . . . 4 |
13 | 12 | a1i 9 | . . 3 |
14 | 2, 13 | rdgon 6345 | . 2 |
15 | 1, 14 | eqeltrd 2241 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 wral 2442 cvv 2721 cmpt 4037 con0 4335 csuc 4337 cfv 5182 (class class class)co 5836 crdg 6328 coa 6372 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-recs 6264 df-irdg 6329 df-oadd 6379 |
This theorem is referenced by: omcl 6420 omv2 6424 |
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