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Theorem ordsucOLD 7835
Description: Obsolete version of ordsuc 7834 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 6391 . . . 4 (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
2 onsuc 7832 . . . . 5 (𝐴 ∈ On → suc 𝐴 ∈ On)
3 eloni 6393 . . . . 5 (suc 𝐴 ∈ On → Ord suc 𝐴)
42, 3syl 17 . . . 4 (𝐴 ∈ On → Ord suc 𝐴)
51, 4biimtrrdi 254 . . 3 (𝐴 ∈ V → (Ord 𝐴 → Ord suc 𝐴))
6 sucidg 6464 . . . 4 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
7 ordelord 6405 . . . . 5 ((Ord suc 𝐴𝐴 ∈ suc 𝐴) → Ord 𝐴)
87ex 412 . . . 4 (Ord suc 𝐴 → (𝐴 ∈ suc 𝐴 → Ord 𝐴))
96, 8syl5com 31 . . 3 (𝐴 ∈ V → (Ord suc 𝐴 → Ord 𝐴))
105, 9impbid 212 . 2 (𝐴 ∈ V → (Ord 𝐴 ↔ Ord suc 𝐴))
11 sucprc 6459 . . . 4 𝐴 ∈ V → suc 𝐴 = 𝐴)
1211eqcomd 2742 . . 3 𝐴 ∈ V → 𝐴 = suc 𝐴)
13 ordeq 6390 . . 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 206   = wceq 1539  wcel 2107  Vcvv 3479  Ord word 6382  Oncon0 6383  suc csuc 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-suc 6389
This theorem is referenced by: (None)
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