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Theorem ordsuc 7801

Description: A class is ordinal if and only if its successor is ordinal. (Contributed by NM, 3-Apr-1995.) Avoid ax-un 7725. (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 7796	. 2 ⊢ (Ord 𝐴 → Ord suc 𝐴)
2		sucidg 6446	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3		ordelord 6387	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → Ord 𝐴)
4	3	ex 414	. . . 4 ⊢ (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴))
5	2, 4	syl5com 31	. . 3 ⊢ (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
6		sucprc 6441	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
7	6	eqcomd 2739	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝐴 = suc 𝐴)
8		ordeq 6372	. . . . 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 247	. . 3 ⊢ (¬ 𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
11	5, 10	pm2.61i 182	. 2 ⊢ (Ord suc 𝐴 → Ord 𝐴)
12	1, 11	impbii 208	1 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3475 Ord word 6364 suc csuc 6367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 df-suc 6371

This theorem is referenced by: ordpwsuc 7803 onsucb 7805 ordsucss 7806 onpsssuc 7807 ordsucelsuc 7810 ordsucsssuc 7811 ordsucuniel 7812 ordsucun 7813 onsucuni2 7822 0elsuc 7823 nlimsucg 7831 limsssuc 7839 cofon1 8671 cofon2 8672 php4 9213 cantnflt 9667 fin23lem26 10320 hsmexlem1 10421 nosupres 27210 noetasuplem4 27239 noetainflem4 27243 scutbdaybnd2lim 27318 satfn 34346 onsuct0 35326 ordsssucim 42153 dfsucon 42274

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