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Theorem limsuc 7217
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 7216 . . 3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
2 suceq 5944 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
32eleq1d 2838 . . . . 5 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
43rspccv 3462 . . . 4 (∀𝑥𝐴 suc 𝑥𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
543ad2ant3 1156 . . 3 ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (𝐵𝐴 → suc 𝐵𝐴))
61, 5sylbi 208 . 2 (Lim 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
7 limord 5938 . . 3 (Lim 𝐴 → Ord 𝐴)
8 ordtr 5891 . . 3 (Ord 𝐴 → Tr 𝐴)
9 trsuc 5964 . . . 4 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
109ex 398 . . 3 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
117, 8, 103syl 18 . 2 (Lim 𝐴 → (suc 𝐵𝐴𝐵𝐴))
126, 11impbid 203 1 (Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  w3a 1098   = wceq 1634  wcel 2148  wral 3064  c0 4073  Tr wtr 4899  Ord word 5876  Lim wlim 5878  suc csuc 5879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-tr 4900  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883
This theorem is referenced by:  limsssuc  7218  limuni3  7220  peano2b  7249  rdgsucg  7693  rdgsucmptnf  7699  oesuclem  7780  oaordi  7801  omordi  7821  oeordi  7842  oelim2  7850  limenpsi  8312  r1tr  8824  r1ordg  8826  r1pwss  8832  r1val1  8834  rankdmr1  8849  rankr1bg  8851  pwwf  8855  rankr1c  8869  rankonidlem  8876  ranklim  8892  r1pwcl  8895  rankxplim3  8929  infxpenlem  9057  alephordi  9118  cflm  9295  cfslb2n  9313  alephreg  9627  r1limwun  9781  rankcf  9822  inatsk  9823
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