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Theorem limsuc 7833
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 7832 . . 3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
2 suceq 6427 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
32eleq1d 2819 . . . . 5 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
43rspccv 3609 . . . 4 (∀𝑥𝐴 suc 𝑥𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
543ad2ant3 1136 . . 3 ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (𝐵𝐴 → suc 𝐵𝐴))
61, 5sylbi 216 . 2 (Lim 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
7 limord 6421 . . 3 (Lim 𝐴 → Ord 𝐴)
8 ordtr 6375 . . 3 (Ord 𝐴 → Tr 𝐴)
9 trsuc 6448 . . . 4 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
109ex 414 . . 3 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
117, 8, 103syl 18 . 2 (Lim 𝐴 → (suc 𝐵𝐴𝐵𝐴))
126, 11impbid 211 1 (Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1088   = wceq 1542  wcel 2107  wral 3062  c0 4321  Tr wtr 5264  Ord word 6360  Lim wlim 6362  suc csuc 6363
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 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
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 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367
This theorem is referenced by:  limsssuc  7834  limuni3  7836  peano2b  7867  rdgsucg  8418  rdgsucmptnf  8424  oesuclem  8520  oaordi  8542  omordi  8562  oeordi  8583  oelim2  8591  limenpsi  9148  r1tr  9767  r1ordg  9769  r1pwss  9775  r1val1  9777  rankdmr1  9792  rankr1bg  9794  pwwf  9798  rankr1c  9812  rankonidlem  9819  ranklim  9835  r1pwcl  9838  rankxplim3  9872  infxpenlem  10004  alephordi  10065  cflm  10241  cfslb2n  10259  alephreg  10573  r1limwun  10727  rankcf  10768  inatsk  10769  oldlim  27361  succlg  42011
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