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Theorem ordsuc 7766

Description: A class is ordinal if and only if its successor is ordinal. (Contributed by NM, 3-Apr-1995.) Avoid ax-un 7690. (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 7763	. 2 ⊢ (Ord 𝐴 → Ord suc 𝐴)
2		sucidg 6408	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3		ordelord 6347	. . . . 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 6403	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
7	6	eqcomd 2743	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝐴 = suc 𝐴)
8		ordeq 6332	. . . . 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 3442 Ord word 6324 suc csuc 6327

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 5243 ax-pr 5379

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 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-tr 5208 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-ord 6328 df-on 6329 df-suc 6331

This theorem is referenced by: ordpwsuc 7767 onsucb 7769 ordsucss 7770 onpsssuc 7771 ordsucelsuc 7774 ordsucsssuc 7775 ordsucuniel 7776 ordsucun 7777 onsucuni2 7786 0elsuc 7787 nlimsucg 7794 limsssuc 7802 cofon1 8610 cofon2 8611 php4 9146 cantnflt 9593 fin23lem26 10247 hsmexlem1 10348 nosupres 27687 noetasuplem4 27716 noetainflem4 27720 cutbdaybnd2lim 27805 satfn 35571 onsuct0 36657 ordsssucim 43759 dfsucon 43879

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