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Mirrors > Home > MPE Home > Th. List > sucon | Structured version Visualization version 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 7772 | . 2 ⊢ ¬ On ∈ V | |
2 | sucprc 6439 | . 2 ⊢ (¬ On ∈ V → suc On = On) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ suc On = On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3469 Oncon0 6363 suc csuc 6365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-ord 6366 df-on 6367 df-suc 6369 |
This theorem is referenced by: ordunisuc 7827 |
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