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Theorem limsuc 7003
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 7002 . . 3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
2 suceq 5754 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
32eleq1d 2683 . . . . 5 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
43rspccv 3295 . . . 4 (∀𝑥𝐴 suc 𝑥𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
543ad2ant3 1082 . . 3 ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (𝐵𝐴 → suc 𝐵𝐴))
61, 5sylbi 207 . 2 (Lim 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
7 limord 5748 . . 3 (Lim 𝐴 → Ord 𝐴)
8 ordtr 5701 . . 3 (Ord 𝐴 → Tr 𝐴)
9 trsuc 5774 . . . 4 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
109ex 450 . . 3 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
117, 8, 103syl 18 . 2 (Lim 𝐴 → (suc 𝐵𝐴𝐵𝐴))
126, 11impbid 202 1 (Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wcel 1987  wral 2907  c0 3896  Tr wtr 4717  Ord word 5686  Lim wlim 5688  suc csuc 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693
This theorem is referenced by:  limsssuc  7004  limuni3  7006  peano2b  7035  rdgsucg  7471  rdgsucmptnf  7477  oesuclem  7557  oaordi  7578  omordi  7598  oeordi  7619  oelim2  7627  limenpsi  8087  r1tr  8591  r1ordg  8593  r1pwss  8599  r1val1  8601  rankdmr1  8616  rankr1bg  8618  pwwf  8622  rankr1c  8636  rankonidlem  8643  ranklim  8659  r1pwcl  8662  rankxplim3  8696  infxpenlem  8788  alephordi  8849  cflm  9024  cfslb2n  9042  alephreg  9356  r1limwun  9510  rankcf  9551  inatsk  9552
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