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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 5719 | . . 3 ⊢ (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴) | |
2 | soinxp 5718 | . . 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 5595 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
5 | df-we 5595 | . 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 205 ∧ wa 397 ∩ cin 3914 Or wor 5549 Fr wfr 5590 We wwe 5592 × cxp 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 |
This theorem is referenced by: wemapwe 9640 infxpenlem 9956 dfac8b 9974 ac10ct 9977 canthwelem 10593 ltbwe 21461 vitali 24993 fin2so 36094 dnwech 41404 aomclem5 41414 |
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