Theorem ordsssucb 44280

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 44347, 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 4049	. 2 ⊢ (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊊ suc 𝐵 ∨ 𝐴 = suc 𝐵))
2		ordsssuc 6443	. . . 4 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
3		eloni 6361	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
4		ordsuci 7805	. . . . 5 ⊢ (Ord 𝐵 → Ord suc 𝐵)
5		ordelpss 6379	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ∈ suc 𝐵 ↔ 𝐴 ⊊ suc 𝐵))
6	3, 4, 5	syl2an 608	. . . 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 930	. 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 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ⊆ wss 3898   ⊊ wpss 3899  Ord word 6350  Oncon0 6351  suc csuc 6353

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6354  df-on 6355  df-suc 6357

This theorem is used by: (None)