Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ordsssucim Structured version   Visualization version   GIF version

Theorem ordsssucim 42832
Description: If an ordinal is less than or equal to the successor of another, then the first is either less than or equal to the second or the first is equal to the successor of the second. Theorem 1 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 See also ordsssucb 42764 for a biimplication when 𝐴 is a set. (Contributed by RP, 3-Jan-2025.)
Assertion
Ref Expression
ordsssucim ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 → (𝐴𝐵𝐴 = suc 𝐵)))

Proof of Theorem ordsssucim
StepHypRef Expression
1 ordsuc 7816 . . 3 (Ord 𝐵 ↔ Ord suc 𝐵)
2 ordsseleq 6398 . . 3 ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ∈ suc 𝐵𝐴 = suc 𝐵)))
31, 2sylan2b 593 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ∈ suc 𝐵𝐴 = suc 𝐵)))
4 simpr 484 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → Ord 𝐵)
5 ordtr 6383 . . . 4 (Ord 𝐵 → Tr 𝐵)
6 trsucss 6457 . . . 4 (Tr 𝐵 → (𝐴 ∈ suc 𝐵𝐴𝐵))
74, 5, 63syl 18 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵𝐴𝐵))
87orim1d 964 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ suc 𝐵𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = suc 𝐵)))
93, 8sylbid 239 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 → (𝐴𝐵𝐴 = suc 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 846   = wceq 1534  wcel 2099  wss 3947  Tr wtr 5265  Ord word 6368  suc csuc 6371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6372  df-on 6373  df-suc 6375
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator