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 7014

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 5731	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
2		suceloni 7013	. . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3		eloni 5733	. . . . 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 5803	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7		ordelord 5745	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → Ord 𝐴)
8	7	ex 450	. . . 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 5800	. . . 4 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
12	11	eqcomd 2628	. . 3 ⊢ (¬ 𝐴 ∈ V → 𝐴 = suc 𝐴)
13		ordeq 5730	. . 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 1483 ∈ wcel 1990 Vcvv 3200 Ord word 5722 Oncon0 5723 suc csuc 5725

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-suc 5729

This theorem is referenced by: ordpwsuc 7015 sucelon 7017 ordsucss 7018 onpsssuc 7019 ordsucelsuc 7022 ordsucsssuc 7023 ordsucuniel 7024 ordsucun 7025 onsucuni2 7034 0elsuc 7035 nlimsucg 7042 limsssuc 7050 php4 8147 cantnflt 8569 fin23lem26 9147 hsmexlem1 9248 nosupres 31853 noetalem3 31865 onsuct0 32440