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

Theorem ordsssucb 43348
Description: An ordinal number is less than or equal to the successor of an ordinal class iff the ordinal number is either less than or equal to the ordinal class or the ordinal number is equal to the successor of the ordinal class. See also ordsssucim 43415, limsssuc 7871. (Contributed by RP, 22-Feb-2025.)
Assertion
Ref Expression
ordsssucb ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴𝐵𝐴 = suc 𝐵)))

Proof of Theorem ordsssucb
StepHypRef Expression
1 sspss 4102 . 2 (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊊ suc 𝐵𝐴 = suc 𝐵))
2 ordsssuc 6473 . . . 4 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵𝐴 ∈ suc 𝐵))
3 eloni 6394 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
4 ordsuci 7828 . . . . 5 (Ord 𝐵 → Ord suc 𝐵)
5 ordelpss 6412 . . . . 5 ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ∈ suc 𝐵𝐴 ⊊ suc 𝐵))
63, 4, 5syl2an 596 . . . 4 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵𝐴 ⊊ suc 𝐵))
72, 6bitrd 279 . . 3 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴𝐵𝐴 ⊊ suc 𝐵))
87orbi1d 917 . 2 ((𝐴 ∈ On ∧ Ord 𝐵) → ((𝐴𝐵𝐴 = suc 𝐵) ↔ (𝐴 ⊊ suc 𝐵𝐴 = suc 𝐵)))
91, 8bitr4id 290 1 ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴𝐵𝐴 = suc 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wss 3951  wpss 3952  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator