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| Mirrors > Home > MPE Home > Th. List > weinxp | Structured version Visualization version GIF version | ||
| Description: Intersection of well-ordering with Cartesian product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.) | 
| Ref | Expression | 
|---|---|
| weinxp | ⊢ (𝑅 We 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | frinxp 5767 | . . 3 ⊢ (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴) | |
| 2 | soinxp 5766 | . . 3 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴) | |
| 3 | 1, 2 | anbi12i 628 | . 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ∧ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)) | 
| 4 | df-we 5638 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
| 5 | df-we 5638 | . 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) We 𝐴 ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ∧ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)) | |
| 6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ (𝑅 We 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∩ cin 3949 Or wor 5590 Fr wfr 5633 We wwe 5635 × cxp 5682 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 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-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 | 
| This theorem is referenced by: wemapwe 9738 infxpenlem 10054 dfac8b 10072 ac10ct 10075 canthwelem 10691 ltbwe 22063 vitali 25649 fin2so 37615 dnwech 43065 aomclem5 43075 | 
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