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Theorem ordsucOLD 7754
Description: Obsolete version of ordsuc 7753 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
StepHypRef Expression
1 elong 6330 . . . 4 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
2 onsuc 7751 . . . . 5 (𝐴 ∈ On → suc 𝐴 ∈ On)
3 eloni 6332 . . . . 5 (suc 𝐴 ∈ On → Ord suc 𝐴)
42, 3syl 17 . . . 4 (𝐴 ∈ On → Ord suc 𝐴)
51, 4syl6bir 254 . . 3 (𝐴 ∈ V → (Ord 𝐴 → Ord suc 𝐴))
6 sucidg 6403 . . . 4 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7 ordelord 6344 . . . . 5 ((Ord suc 𝐴𝐴 ∈ suc 𝐴) → Ord 𝐴)
87ex 414 . . . 4 (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴))
96, 8syl5com 31 . . 3 (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
105, 9impbid 211 . 2 (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
11 sucprc 6398 . . . 4 𝐴 ∈ V → suc 𝐴 = 𝐴)
1211eqcomd 2743 . . 3 𝐴 ∈ V → 𝐴 = suc 𝐴)
13 ordeq 6329 . . 3 (𝐴 = suc 𝐴 → (Ord 𝐴 ↔ Ord suc 𝐴))
1412, 13syl 17 . 2 𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
1510, 14pm2.61i 182 1 (Ord 𝐴 ↔ Ord suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1542  wcel 2107  Vcvv 3448  Ord word 6321  Oncon0 6322  suc csuc 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by: (None)
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