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Theorem limsuc 7555
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 7554 . . 3 (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
2 suceq 6253 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
32eleq1d 2901 . . . . 5 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
43rspccv 3623 . . . 4 (∀𝑥𝐴 suc 𝑥𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
543ad2ant3 1129 . . 3 ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → (𝐵𝐴 → suc 𝐵𝐴))
61, 5sylbi 218 . 2 (Lim 𝐴 → (𝐵𝐴 → suc 𝐵𝐴))
7 limord 6247 . . 3 (Lim 𝐴 → Ord 𝐴)
8 ordtr 6202 . . 3 (Ord 𝐴 → Tr 𝐴)
9 trsuc 6272 . . . 4 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
109ex 413 . . 3 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
117, 8, 103syl 18 . 2 (Lim 𝐴 → (suc 𝐵𝐴𝐵𝐴))
126, 11impbid 213 1 (Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1081   = wceq 1530  wcel 2107  wral 3142  c0 4294  Tr wtr 5168  Ord word 6187  Lim wlim 6189  suc csuc 6190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-tr 5169  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194
This theorem is referenced by:  limsssuc  7556  limuni3  7558  peano2b  7587  rdgsucg  8053  rdgsucmptnf  8059  oesuclem  8144  oaordi  8165  omordi  8185  oeordi  8206  oelim2  8214  limenpsi  8684  r1tr  9197  r1ordg  9199  r1pwss  9205  r1val1  9207  rankdmr1  9222  rankr1bg  9224  pwwf  9228  rankr1c  9242  rankonidlem  9249  ranklim  9265  r1pwcl  9268  rankxplim3  9302  infxpenlem  9431  alephordi  9492  cflm  9664  cfslb2n  9682  alephreg  9996  r1limwun  10150  rankcf  10191  inatsk  10192
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