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Theorem ordsssucim 43415
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 43348 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 7833 . . 3 (Ord 𝐵 ↔ Ord suc 𝐵)
2 ordsseleq 6413 . . 3 ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ∈ suc 𝐵𝐴 = suc 𝐵)))
31, 2sylan2b 594 . 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 6398 . . . 4 (Ord 𝐵 → Tr 𝐵)
6 trsucss 6472 . . . 4 (Tr 𝐵 → (𝐴 ∈ suc 𝐵𝐴𝐵))
74, 5, 63syl 18 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵𝐴𝐵))
87orim1d 968 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ suc 𝐵𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = suc 𝐵)))
93, 8sylbid 240 1 ((Ord 𝐴 ∧ 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  Tr wtr 5259  Ord word 6383  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)
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