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Theorem ordsuc 7807

Description: A class is ordinal if and only if its successor is ordinal. (Contributed by NM, 3-Apr-1995.) Avoid ax-un 7729. (Revised by BTernaryTau, 6-Jan-2025.)

**Assertion**
Ref	Expression
ordsuc	⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

**Proof of Theorem ordsuc**
Step	Hyp	Ref	Expression
1		ordsuci 7802	. 2 ⊢ (Ord 𝐴 → Ord suc 𝐴)
2		sucidg 6435	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3		ordelord 6374	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → Ord 𝐴)
4	3	ex 412	. . . 4 ⊢ (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴))
5	2, 4	syl5com 31	. . 3 ⊢ (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
6		sucprc 6430	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
7	6	eqcomd 2741	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝐴 = suc 𝐴)
8		ordeq 6359	. . . . 5 ⊢ (𝐴 = suc 𝐴 → (Ord 𝐴 ↔ Ord suc 𝐴))
9	7, 8	syl 17	. . . 4 ⊢ (¬ 𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
10	9	biimprd 248	. . 3 ⊢ (¬ 𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
11	5, 10	pm2.61i 182	. 2 ⊢ (Ord suc 𝐴 → Ord 𝐴)
12	1, 11	impbii 209	1 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3459 Ord word 6351 suc csuc 6354

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 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2727 df-clel 2809 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-tr 5230 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-ord 6355 df-on 6356 df-suc 6358

This theorem is referenced by: ordpwsuc 7809 onsucb 7811 ordsucss 7812 onpsssuc 7813 ordsucelsuc 7816 ordsucsssuc 7817 ordsucuniel 7818 ordsucun 7819 onsucuni2 7828 0elsuc 7829 nlimsucg 7837 limsssuc 7845 cofon1 8684 cofon2 8685 php4 9224 cantnflt 9686 fin23lem26 10339 hsmexlem1 10440 nosupres 27671 noetasuplem4 27700 noetainflem4 27704 scutbdaybnd2lim 27781 satfn 35377 onsuct0 36459 ordsssucim 43426 dfsucon 43547

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