|   | Mathbox for Richard Penner | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > ordsssucb | Structured version Visualization version GIF version | ||
| 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 43415, limsssuc 7871. (Contributed by RP, 22-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| ordsssucb | ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊆ 𝐵 ∨ 𝐴 = suc 𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sspss 4102 | . 2 ⊢ (𝐴 ⊆ suc 𝐵 ↔ (𝐴 ⊊ suc 𝐵 ∨ 𝐴 = suc 𝐵)) | |
| 2 | ordsssuc 6473 | . . . 4 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)) | |
| 3 | eloni 6394 | . . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 4 | ordsuci 7828 | . . . . 5 ⊢ (Ord 𝐵 → Ord suc 𝐵) | |
| 5 | ordelpss 6412 | . . . . 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 917 | . 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))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ⊊ wpss 3952 Ord word 6383 Oncon0 6384 suc csuc 6386 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |