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Theorem ordsuc 7531

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 6201	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
2		suceloni 7530	. . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3		eloni 6203	. . . . 5 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴)
4	2, 3	syl 17	. . . 4 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
5	1, 4	syl6bir 256	. . 3 ⊢ (𝐴 ∈ V → (Ord 𝐴 → Ord suc 𝐴))
6		sucidg 6271	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7		ordelord 6215	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → Ord 𝐴)
8	7	ex 415	. . . 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 214	. 2 ⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
11		sucprc 6268	. . . 4 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
12	11	eqcomd 2829	. . 3 ⊢ (¬ 𝐴 ∈ V → 𝐴 = suc 𝐴)
13		ordeq 6200	. . 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 184	1 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3496 Ord word 6192 Oncon0 6193 suc csuc 6195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 df-suc 6199

This theorem is referenced by: ordpwsuc 7532 sucelon 7534 ordsucss 7535 onpsssuc 7536 ordsucelsuc 7539 ordsucsssuc 7540 ordsucuniel 7541 ordsucun 7542 onsucuni2 7551 0elsuc 7552 nlimsucg 7559 limsssuc 7567 php4 8706 cantnflt 9137 fin23lem26 9749 hsmexlem1 9850 satfn 32604 nosupres 33209 noetalem3 33221 onsuct0 33791 dfsucon 39896