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Theorem ordsucOLD 7815
Description: Obsolete version of ordsuc 7814 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 6372 . . . 4 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
2 onsuc 7812 . . . . 5 (𝐴 ∈ On → suc 𝐴 ∈ On)
3 eloni 6374 . . . . 5 (suc 𝐴 ∈ On → Ord suc 𝐴)
42, 3syl 17 . . . 4 (𝐴 ∈ On → Ord suc 𝐴)
51, 4syl6bir 253 . . 3 (𝐴 ∈ V → (Ord 𝐴 → Ord suc 𝐴))
6 sucidg 6445 . . . 4 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7 ordelord 6386 . . . . 5 ((Ord suc 𝐴𝐴 ∈ suc 𝐴) → Ord 𝐴)
87ex 411 . . . 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 6440 . . . 4 𝐴 ∈ V → suc 𝐴 = 𝐴)
1211eqcomd 2731 . . 3 𝐴 ∈ V → 𝐴 = suc 𝐴)
13 ordeq 6371 . . 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 1533  wcel 2098  Vcvv 3463  Ord word 6363  Oncon0 6364  suc csuc 6366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368  df-suc 6370
This theorem is referenced by: (None)
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