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Theorem ordsuc 7758

Description: A class is ordinal if and only if its successor is ordinal. (Contributed by NM, 3-Apr-1995.) Avoid ax-un 7682. (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 7755	. 2 ⊢ (Ord 𝐴 → Ord suc 𝐴)
2		sucidg 6400	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3		ordelord 6339	. . . . 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 6395	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
7	6	eqcomd 2743	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝐴 = suc 𝐴)
8		ordeq 6324	. . . . 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 1542 ∈ wcel 2114 Vcvv 3430 Ord word 6316 suc csuc 6319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2709 ax-sep 5231 ax-pr 5370

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2716 df-cleq 2729 df-clel 2812 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-tr 5194 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-ord 6320 df-on 6321 df-suc 6323

This theorem is referenced by: ordpwsuc 7759 onsucb 7761 ordsucss 7762 onpsssuc 7763 ordsucelsuc 7766 ordsucsssuc 7767 ordsucuniel 7768 ordsucun 7769 onsucuni2 7778 0elsuc 7779 nlimsucg 7786 limsssuc 7794 cofon1 8601 cofon2 8602 php4 9137 cantnflt 9584 fin23lem26 10238 hsmexlem1 10339 nosupres 27685 noetasuplem4 27714 noetainflem4 27718 cutbdaybnd2lim 27803 satfn 35553 onsuct0 36639 ordsssucim 43848 dfsucon 43968

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