Theorem ordsssucb 44090

Description: An ordinal number is less than or equal to the successor of an ordinal class iff the ordinal number is either less than or equal to the ordinal class or the ordinal number is equal to the successor of the ordinal class. See also ordsssucim 44157, limsssuc 7844. (Contributed by RP, 22-Feb-2025.)

Assertion

Ref	Expression
ordsssucb	⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))

Proof of Theorem ordsssucb

Step	Hyp	Ref	Expression
1		sspss 4055	. 2 ⊢ (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊊ suc 𝐵 ∨ 𝐴 = suc 𝐵))
2		ordsssuc 6452	. . . 4 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
3		eloni 6370	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
4		ordsuci 7805	. . . . 5 ⊢ (Ord 𝐵 → Ord suc 𝐵)
5		ordelpss 6388	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ∈ suc 𝐵 ↔ 𝐴 ⊊ suc 𝐵))
6	3, 4, 5	syl2an 607	. . . 4 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 ↔ 𝐴 ⊊ suc 𝐵))
7	2, 6	bitrd 282	. . 3 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊊ suc 𝐵))
8	7	orbi1d 929	. 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → ((𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵) ↔ (𝐴 ⊊ suc 𝐵 ∨ 𝐴 = suc 𝐵)))
9	1, 8	bitr4id 293	1 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1569   ∈ wcel 2142   ⊆ wss 3904   ⊊ wpss 3905  Ord word 6359  Oncon0 6360  suc csuc 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-tr 5218  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-ord 6363  df-on 6364  df-suc 6366

This theorem is used by: (None)