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Mirrors > Home > ILE Home > Th. List > sucon | GIF version |
Description: The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.) |
Ref | Expression |
---|---|
sucon | ⊢ suc On = On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onprc 4295 | . 2 ⊢ ¬ On ∈ V | |
2 | sucprc 4167 | . 2 ⊢ (¬ On ∈ V → suc On = On) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ suc On = On |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1284 ∈ wcel 1433 Vcvv 2601 Oncon0 4118 suc csuc 4120 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 |
This theorem is referenced by: (None) |
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