Theorem ordsssucb 44529

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 44596, limsssuc 7885. (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 4032	. 2 ⊢ (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊊ suc 𝐵 ∨ 𝐴 = suc 𝐵))
2		ordsssuc 6472	. . . 4 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
3		eloni 6390	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
4		ordsuci 7846	. . . . 5 ⊢ (Ord 𝐵 → Ord suc 𝐵)
5		ordelpss 6408	. . . . 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 2147   ⊆ wss 3900   ⊊ wpss 4016  Ord word 6379  Oncon0 6380  suc csuc 6382

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  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 1573  df-fal 1583  df-ex 1813  df-sb 2102  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 6383  df-on 6384  df-suc 6386

This theorem is used by: (None)