Theorem ordsssucim 44547

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 44480 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 7835	. . 3 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
2		ordsseleq 6399	. . 3 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ∈ suc 𝐵 ∨ 𝐴 = suc 𝐵)))
3	1, 2	sylan2b 606	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ∈ suc 𝐵 ∨ 𝐴 = suc 𝐵)))
4		simpr 490	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord 𝐵)
5		ordtr 6383	. . . 4 ⊢ (Ord 𝐵 → Tr 𝐵)
6		trsucss 6460	. . . 4 ⊢ (Tr 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵))
7	4, 5, 6	3syl 19	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵))
8	7	orim1d 981	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ suc 𝐵 ∨ 𝐴 = suc 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))
9	3, 8	sylbid 243	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 → (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2146   ⊆ wss 3900  Tr wtr 5215  Ord word 6368  suc csuc 6371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6372  df-on 6373  df-suc 6375

This theorem is used by: (None)