Theorem ordsssucim 43385

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 43318 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 → (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))

Proof of Theorem ordsssucim

Step	Hyp	Ref	Expression
1		ordsuc 7747	. . 3 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
2		ordsseleq 6336	. . 3 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ∈ suc 𝐵 ∨ 𝐴 = suc 𝐵)))
3	1, 2	sylan2b 594	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ∈ suc 𝐵 ∨ 𝐴 = suc 𝐵)))
4		simpr 484	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord 𝐵)
5		ordtr 6321	. . . 4 ⊢ (Ord 𝐵 → Tr 𝐵)
6		trsucss 6397	. . . 4 ⊢ (Tr 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵))
7	4, 5, 6	3syl 18	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵))
8	7	orim1d 967	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ suc 𝐵 ∨ 𝐴 = suc 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))
9	3, 8	sylbid 240	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ⊆ wss 3903  Tr wtr 5199  Ord word 6306  suc csuc 6309

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-tr 5200  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6310  df-on 6311  df-suc 6313

This theorem is referenced by: (None)