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Theorem ordsuc 7165

Description: The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.)

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

**Proof of Theorem ordsuc**
Step	Hyp	Ref	Expression
1		elong 5873	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
2		suceloni 7164	. . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3		eloni 5875	. . . . 5 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴)
4	2, 3	syl 17	. . . 4 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
5	1, 4	syl6bir 244	. . 3 ⊢ (𝐴 ∈ V → (Ord 𝐴 → Ord suc 𝐴))
6		sucidg 5945	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7		ordelord 5887	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → Ord 𝐴)
8	7	ex 397	. . . 4 ⊢ (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴))
9	6, 8	syl5com 31	. . 3 ⊢ (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
10	5, 9	impbid 202	. 2 ⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
11		sucprc 5942	. . . 4 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
12	11	eqcomd 2777	. . 3 ⊢ (¬ 𝐴 ∈ V → 𝐴 = suc 𝐴)
13		ordeq 5872	. . 3 ⊢ (𝐴 = suc 𝐴 → (Ord 𝐴 ↔ Ord suc 𝐴))
14	12, 13	syl 17	. 2 ⊢ (¬ 𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
15	10, 14	pm2.61i 176	1 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1631 ∈ wcel 2145 Vcvv 3351 Ord word 5864 Oncon0 5865 suc csuc 5867

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 ax-un 7100

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-tr 4888 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-ord 5868 df-on 5869 df-suc 5871

This theorem is referenced by: ordpwsuc 7166 sucelon 7168 ordsucss 7169 onpsssuc 7170 ordsucelsuc 7173 ordsucsssuc 7174 ordsucuniel 7175 ordsucun 7176 onsucuni2 7185 0elsuc 7186 nlimsucg 7193 limsssuc 7201 php4 8307 cantnflt 8737 fin23lem26 9353 hsmexlem1 9454 nosupres 32190 noetalem3 32202 onsuct0 32777