ordsuc - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ordsuc		Structured version Visualization version GIF version

Theorem ordsuc 7747

Description: A class is ordinal if and only if its successor is ordinal. (Contributed by NM, 3-Apr-1995.) Avoid ax-un 7671. (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 7742	. 2 ⊢ (Ord 𝐴 → Ord suc 𝐴)
2		sucidg 6398	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3		ordelord 6339	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → Ord 𝐴)
4	3	ex 413	. . . 4 ⊢ (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴))
5	2, 4	syl5com 31	. . 3 ⊢ (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
6		sucprc 6393	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
7	6	eqcomd 2742	. . . . 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 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 1541 ∈ wcel 2106 Vcvv 3445 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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-tr 5223 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-ord 6320 df-on 6321 df-suc 6323

This theorem is referenced by: ordpwsuc 7749 onsucb 7751 ordsucss 7752 onpsssuc 7753 ordsucelsuc 7756 ordsucsssuc 7757 ordsucuniel 7758 ordsucun 7759 onsucuni2 7768 0elsuc 7769 nlimsucg 7777 limsssuc 7785 cofon1 8617 cofon2 8618 php4 9156 cantnflt 9607 fin23lem26 10260 hsmexlem1 10361 nosupres 27053 noetasuplem4 27082 noetainflem4 27086 scutbdaybnd2lim 27154 satfn 33940 onsuct0 34904 dfsucon 41777

W3C validator