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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 2106 Oncon0 6386 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-suc 6392 |
This theorem is referenced by: 1onOLD 8518 2onOLD 8520 3on 8523 4on 8524 tz9.12lem2 9826 tz9.12 9828 rankpwi 9861 bndrank 9879 rankval4 9905 rankmapu 9916 rankxplim3 9919 cfcof 10312 ttukeylem6 10552 n0sbday 28369 pw2bday 28433 onsucconni 36420 onsucsuccmpi 36426 |
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