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Theorem limsuc2 43053
Description: Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Assertion
Ref Expression
limsuc2 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))

Proof of Theorem limsuc2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ordunisuc2 7865 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
21biimpa 476 . . . 4 ((Ord 𝐴𝐴 = 𝐴) → ∀𝑥𝐴 suc 𝑥𝐴)
3 suceq 6450 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
43eleq1d 2826 . . . . 5 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
54rspccva 3621 . . . 4 ((∀𝑥𝐴 suc 𝑥𝐴𝐵𝐴) → suc 𝐵𝐴)
62, 5sylan 580 . . 3 (((Ord 𝐴𝐴 = 𝐴) ∧ 𝐵𝐴) → suc 𝐵𝐴)
76ex 412 . 2 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 → suc 𝐵𝐴))
8 ordtr 6398 . . . 4 (Ord 𝐴 → Tr 𝐴)
9 trsuc 6471 . . . . 5 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
109ex 412 . . . 4 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
118, 10syl 17 . . 3 (Ord 𝐴 → (suc 𝐵𝐴𝐵𝐴))
1211adantr 480 . 2 ((Ord 𝐴𝐴 = 𝐴) → (suc 𝐵𝐴𝐵𝐴))
137, 12impbid 212 1 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061   cuni 4907  Tr wtr 5259  Ord word 6383  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  aomclem4  43069
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