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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 4511 | . 2 ⊢ ¬ On ∈ V | |
2 | sucprc 4372 | . 2 ⊢ (¬ On ∈ V → suc On = On) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ suc On = On |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1335 ∈ wcel 2128 Vcvv 2712 Oncon0 4323 suc 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-in1 604 ax-in2 605 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-ext 2139 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-sn 3566 df-uni 3773 df-tr 4063 df-iord 4326 df-on 4328 df-suc 4331 |
This theorem is referenced by: (None) |
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