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| Mirrors > Home > MPE Home > Th. List > onsuci | Structured version Visualization version GIF version | ||
| Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7831 and onsucb 7837. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.) |
| Ref | Expression |
|---|---|
| onssi.1 | ⊢ 𝐴 ∈ On |
| Ref | Expression |
|---|---|
| onsuci | ⊢ suc 𝐴 ∈ On |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | onssi.1 | . 2 ⊢ 𝐴 ∈ On | |
| 2 | onsuc 7831 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ suc 𝐴 ∈ On |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 Oncon0 6384 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 |
| This theorem is referenced by: 1onOLD 8519 2onOLD 8521 3on 8524 4on 8525 tz9.12lem2 9828 tz9.12 9830 rankpwi 9863 bndrank 9881 rankval4 9907 rankmapu 9918 rankxplim3 9921 cfcof 10314 ttukeylem6 10554 n0sbday 28354 pw2bday 28418 onsucconni 36438 onsucsuccmpi 36444 |
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