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| Mirrors > Home > MPE Home > Th. List > onsuc | Structured version Visualization version GIF version | ||
| Description: The successor of an ordinal number is an ordinal number. Closed form of onsuci 7859. Forward implication of onsucb 7837. Proposition 7.24 of [TakeutiZaring] p. 41. Remark 1.5 of [Schloeder] p. 1. (Contributed by NM, 6-Jun-1994.) (Proof shortened by BTernaryTau, 30-Nov-2024.) |
| Ref | Expression |
|---|---|
| onsuc | ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sucexg 7825 | . 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V) | |
| 2 | sucexeloni 7829 | . 2 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On) | |
| 3 | 1, 2 | mpdan 687 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 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: ordsucOLD 7834 unon 7851 onsuci 7859 ordunisuc2 7865 ordzsl 7866 onzsl 7867 tfindsg 7882 dfom2 7889 findsg 7919 tfrlem12 8429 oasuc 8562 omsuc 8564 onasuc 8566 oacl 8573 oneo 8619 omeulem1 8620 omeulem2 8621 oeordi 8625 oeworde 8631 oelim2 8633 oelimcl 8638 oeeulem 8639 oeeui 8640 oaabs2 8687 naddsuc2 8739 omxpenlem 9113 card2inf 9595 cantnflt 9712 cantnflem1d 9728 cnfcom 9740 r1ordg 9818 bndrank 9881 r1pw 9885 r1pwALT 9886 tcrank 9924 onssnum 10080 dfac12lem2 10185 cfsuc 10297 cfsmolem 10310 fin1a2lem1 10440 fin1a2lem2 10441 ttukeylem7 10555 alephreg 10622 gch2 10715 winainflem 10733 winalim2 10736 r1wunlim 10777 nqereu 10969 noextend 27711 noresle 27742 nosupno 27748 madeoldsuc 27923 constrextdg2lem 33789 ontgval 36432 ontgsucval 36433 onsuctop 36434 sucneqond 37366 onexgt 43252 onexomgt 43253 onexoegt 43256 onepsuc 43264 onsucelab 43276 ordnexbtwnsuc 43280 onsucrn 43284 cantnftermord 43333 cantnfub2 43335 omabs2 43345 onsucunipr 43385 onsucunitp 43386 nadd1suc 43405 naddwordnexlem0 43409 naddwordnexlem1 43410 minregex 43547 onsetreclem2 49225 |
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