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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 7571 | . 2 ⊢ ¬ On ∈ V | |
2 | sucprc 6297 | . 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 1543 ∈ wcel 2111 Vcvv 3415 Oncon0 6222 suc csuc 6224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-11 2159 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pr 5331 ax-un 7532 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3417 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-br 5063 df-opab 5125 df-tr 5171 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-we 5520 df-ord 6225 df-on 6226 df-suc 6228 |
This theorem is referenced by: ordunisuc 7620 |
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