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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 5771 | . . 3 ⊢ (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴) | |
2 | soinxp 5770 | . . 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 5643 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
5 | df-we 5643 | . 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 3962 Or wor 5596 Fr wfr 5638 We wwe 5640 × cxp 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 |
This theorem is referenced by: wemapwe 9735 infxpenlem 10051 dfac8b 10069 ac10ct 10072 canthwelem 10688 ltbwe 22080 vitali 25662 fin2so 37594 dnwech 43037 aomclem5 43047 |
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