Theorem ordsucOLD 7834

Description: Obsolete version of ordsuc 7833 as of 6-Jan-2025. (Contributed by NM, 3-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ordsucOLD	⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

Proof of Theorem ordsucOLD

Step	Hyp	Ref	Expression
1		elong 6394	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
2		onsuc 7831	. . . . 5 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
3		eloni 6396	. . . . 5 ⊢ (suc 𝐴 ∈ On → Ord suc 𝐴)
4	2, 3	syl 17	. . . 4 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
5	1, 4	biimtrrdi 254	. . 3 ⊢ (𝐴 ∈ V → (Ord 𝐴 → Ord suc 𝐴))
6		sucidg 6467	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7		ordelord 6408	. . . . 5 ⊢ ((Ord suc 𝐴 ∧ 𝐴 ∈ suc 𝐴) → Ord 𝐴)
8	7	ex 412	. . . 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 212	. 2 ⊢ (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
11		sucprc 6462	. . . 4 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
12	11	eqcomd 2741	. . 3 ⊢ (¬ 𝐴 ∈ V → 𝐴 = suc 𝐴)
13		ordeq 6393	. . 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 182	1 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ∈ wcel 2106  Vcvv 3478  Ord word 6385  Oncon0 6386  suc csuc 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392

This theorem is referenced by: (None)