Mathbox for Stefan O'Rear < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  limsuc2 Structured version   Visualization version   GIF version

Theorem limsuc2 36629
 Description: Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Assertion
Ref Expression
limsuc2 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))

Proof of Theorem limsuc2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ordunisuc2 6936 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
21biimpa 500 . . . 4 ((Ord 𝐴𝐴 = 𝐴) → ∀𝑥𝐴 suc 𝑥𝐴)
3 suceq 5707 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
43eleq1d 2672 . . . . 5 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
54rspccva 3281 . . . 4 ((∀𝑥𝐴 suc 𝑥𝐴𝐵𝐴) → suc 𝐵𝐴)
62, 5sylan 487 . . 3 (((Ord 𝐴𝐴 = 𝐴) ∧ 𝐵𝐴) → suc 𝐵𝐴)
76ex 449 . 2 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 → suc 𝐵𝐴))
8 ordtr 5654 . . . 4 (Ord 𝐴 → Tr 𝐴)
9 trsuc 5727 . . . . 5 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
109ex 449 . . . 4 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
118, 10syl 17 . . 3 (Ord 𝐴 → (suc 𝐵𝐴𝐵𝐴))
1211adantr 480 . 2 ((Ord 𝐴𝐴 = 𝐴) → (suc 𝐵𝐴𝐵𝐴))
137, 12impbid 201 1 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∪ cuni 4372  Tr wtr 4680  Ord word 5639  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646 This theorem is referenced by:  aomclem4  36645
 Copyright terms: Public domain W3C validator