Theorem ordsssucim 43852

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 43785 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 7760	. . 3 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
2		ordsseleq 6348	. . 3 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ∈ suc 𝐵 ∨ 𝐴 = suc 𝐵)))
3	1, 2	sylan2b 595	. 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 6333	. . . 4 ⊢ (Ord 𝐵 → Tr 𝐵)
6		trsucss 6409	. . . 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 968	. 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 848   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  Tr wtr 5193  Ord word 6318  suc csuc 6321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-ord 6322  df-on 6323  df-suc 6325

This theorem is referenced by: (None)