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Theorem limsuc 7870
Description: The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
limsuc (Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))

Proof of Theorem limsuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dflim4 7869 . . 3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
2 suceq 6450 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
32eleq1d 2826 . . . . 5 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
43rspccv 3619 . . . 4 (∀𝑥𝐴 suc 𝑥𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
543ad2ant3 1136 . . 3 ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (𝐵𝐴 → suc 𝐵𝐴))
61, 5sylbi 217 . 2 (Lim 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
7 limord 6444 . . 3 (Lim 𝐴 → Ord 𝐴)
8 ordtr 6398 . . 3 (Ord 𝐴 → Tr 𝐴)
9 trsuc 6471 . . . 4 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
109ex 412 . . 3 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
117, 8, 103syl 18 . 2 (Lim 𝐴 → (suc 𝐵𝐴𝐵𝐴))
126, 11impbid 212 1 (Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  wral 3061  c0 4333  Tr wtr 5259  Ord word 6383  Lim wlim 6385  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-lim 6389  df-suc 6390
This theorem is referenced by:  limsssuc  7871  limuni3  7873  peano2b  7904  rdgsucg  8463  rdgsucmptnf  8469  oesuclem  8563  oaordi  8584  omordi  8604  oeordi  8625  oelim2  8633  limenpsi  9192  r1tr  9816  r1ordg  9818  r1pwss  9824  r1val1  9826  rankdmr1  9841  rankr1bg  9843  pwwf  9847  rankr1c  9861  rankonidlem  9868  ranklim  9884  r1pwcl  9887  rankxplim3  9921  infxpenlem  10053  alephordi  10114  cflm  10290  cfslb2n  10308  alephreg  10622  r1limwun  10776  rankcf  10817  inatsk  10818  oldlim  27925  succlg  43341
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