![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > oneptri | Structured version Visualization version GIF version |
Description: The strict, complete (linear) order on the ordinals is complete. Theorem 1.9(iii) of [Schloeder] p. 1. (Contributed by RP, 15-Jan-2025.) |
Ref | Expression |
---|---|
oneptri | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epsoon 43243 | . . 3 ⊢ E Or On | |
2 | sotrieq 5621 | . . 3 ⊢ (( E Or On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 = 𝐵 ↔ ¬ (𝐴 E 𝐵 ∨ 𝐵 E 𝐴))) | |
3 | 1, 2 | mpan 690 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 = 𝐵 ↔ ¬ (𝐴 E 𝐵 ∨ 𝐵 E 𝐴))) |
4 | xoror 1518 | . . 3 ⊢ (((𝐴 E 𝐵 ∨ 𝐵 E 𝐴) ⊻ 𝐴 = 𝐵) → ((𝐴 E 𝐵 ∨ 𝐵 E 𝐴) ∨ 𝐴 = 𝐵)) | |
5 | xorcom 1514 | . . . 4 ⊢ (((𝐴 E 𝐵 ∨ 𝐵 E 𝐴) ⊻ 𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ⊻ (𝐴 E 𝐵 ∨ 𝐵 E 𝐴))) | |
6 | df-xor 1512 | . . . 4 ⊢ ((𝐴 = 𝐵 ⊻ (𝐴 E 𝐵 ∨ 𝐵 E 𝐴)) ↔ ¬ (𝐴 = 𝐵 ↔ (𝐴 E 𝐵 ∨ 𝐵 E 𝐴))) | |
7 | xor3 382 | . . . 4 ⊢ (¬ (𝐴 = 𝐵 ↔ (𝐴 E 𝐵 ∨ 𝐵 E 𝐴)) ↔ (𝐴 = 𝐵 ↔ ¬ (𝐴 E 𝐵 ∨ 𝐵 E 𝐴))) | |
8 | 5, 6, 7 | 3bitrri 298 | . . 3 ⊢ ((𝐴 = 𝐵 ↔ ¬ (𝐴 E 𝐵 ∨ 𝐵 E 𝐴)) ↔ ((𝐴 E 𝐵 ∨ 𝐵 E 𝐴) ⊻ 𝐴 = 𝐵)) |
9 | df-3or 1088 | . . 3 ⊢ ((𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵) ↔ ((𝐴 E 𝐵 ∨ 𝐵 E 𝐴) ∨ 𝐴 = 𝐵)) | |
10 | 4, 8, 9 | 3imtr4i 292 | . 2 ⊢ ((𝐴 = 𝐵 ↔ ¬ (𝐴 E 𝐵 ∨ 𝐵 E 𝐴)) → (𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵)) |
11 | 3, 10 | syl 17 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵 ∨ 𝐵 E 𝐴 ∨ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 ⊻ wxo 1511 = wceq 1540 ∈ wcel 2108 class class class wbr 5141 E cep 5581 Or wor 5589 Oncon0 6382 |
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 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-tr 5258 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-ord 6385 df-on 6386 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |