Theorem ordsssucim 42897

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 42829 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 7814	. . 3 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
2		ordsseleq 6393	. . 3 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ∈ suc 𝐵 ∨ 𝐴 = suc 𝐵)))
3	1, 2	sylan2b 592	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ∈ suc 𝐵 ∨ 𝐴 = suc 𝐵)))
4		simpr 483	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord 𝐵)
5		ordtr 6378	. . . 4 ⊢ (Ord 𝐵 → Tr 𝐵)
6		trsucss 6452	. . . 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 963	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ suc 𝐵 ∨ 𝐴 = suc 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))
9	3, 8	sylbid 239	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098   ⊆ wss 3939  Tr wtr 5260  Ord word 6363  suc csuc 6366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368  df-suc 6370

This theorem is referenced by: (None)