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Theorem limsuc2 41397
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 7785 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
21biimpa 478 . . . 4 ((Ord 𝐴𝐴 = 𝐴) → ∀𝑥𝐴 suc 𝑥𝐴)
3 suceq 6388 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
43eleq1d 2823 . . . . 5 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
54rspccva 3583 . . . 4 ((∀𝑥𝐴 suc 𝑥𝐴𝐵𝐴) → suc 𝐵𝐴)
62, 5sylan 581 . . 3 (((Ord 𝐴𝐴 = 𝐴) ∧ 𝐵𝐴) → suc 𝐵𝐴)
76ex 414 . 2 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 → suc 𝐵𝐴))
8 ordtr 6336 . . . 4 (Ord 𝐴 → Tr 𝐴)
9 trsuc 6409 . . . . 5 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
109ex 414 . . . 4 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
118, 10syl 17 . . 3 (Ord 𝐴 → (suc 𝐵𝐴𝐵𝐴))
1211adantr 482 . 2 ((Ord 𝐴𝐴 = 𝐴) → (suc 𝐵𝐴𝐵𝐴))
137, 12impbid 211 1 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065   cuni 4870  Tr wtr 5227  Ord word 6321  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:  aomclem4  41413
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