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| Mirrors > Home > MPE Home > Th. List > soinxp | Structured version Visualization version GIF version | ||
| Description: Intersection of total order with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.) | 
| Ref | Expression | 
|---|---|
| soinxp | ⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | poinxp 5766 | . . 3 ⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴) | |
| 2 | brinxp 5764 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦)) | |
| 3 | biidd 262 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)) | |
| 4 | brinxp 5764 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | |
| 5 | 4 | ancoms 458 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | 
| 6 | 2, 3, 5 | 3orbi123d 1437 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))) | 
| 7 | 6 | ralbidva 3176 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))) | 
| 8 | 7 | ralbiia 3091 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)) | 
| 9 | 1, 8 | anbi12i 628 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))) | 
| 10 | df-so 5593 | . 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | |
| 11 | df-so 5593 | . 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴 ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))) | |
| 12 | 9, 10, 11 | 3bitr4i 303 | 1 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∨ w3o 1086 ∈ wcel 2108 ∀wral 3061 ∩ cin 3950 class class class wbr 5143 Po wpo 5590 Or wor 5591 × cxp 5683 | 
| 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-po 5592 df-so 5593 df-xp 5691 | 
| This theorem is referenced by: weinxp 5770 ltsopi 10928 cnso 16283 opsrtoslem2 22080 | 
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