Theorem ordsssucb 43326

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 43393, limsssuc 7850. (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 4082	. 2 ⊢ (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊊ suc 𝐵 ∨ 𝐴 = suc 𝐵))
2		ordsssuc 6448	. . . 4 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
3		eloni 6367	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
4		ordsuci 7807	. . . . 5 ⊢ (Ord 𝐵 → Ord suc 𝐵)
5		ordelpss 6385	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ∈ suc 𝐵 ↔ 𝐴 ⊊ suc 𝐵))
6	3, 4, 5	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 ↔ 𝐴 ⊊ suc 𝐵))
7	2, 6	bitrd 279	. . 3 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊊ suc 𝐵))
8	7	orbi1d 916	. 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → ((𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵) ↔ (𝐴 ⊊ suc 𝐵 ∨ 𝐴 = suc 𝐵)))
9	1, 8	bitr4id 290	1 ⊢ ((𝐴 ∈ On ∧ 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 3931   ⊊ wpss 3932  Ord word 6356  Oncon0 6357  suc csuc 6359

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361  df-suc 6363

This theorem is referenced by: (None)