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Mirrors > Home > MPE Home > Th. List > elin2 | Structured version Visualization version GIF version |
Description: Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
elin2.x | ⊢ 𝑋 = (𝐵 ∩ 𝐶) |
Ref | Expression |
---|---|
elin2 | ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin2.x | . . 3 ⊢ 𝑋 = (𝐵 ∩ 𝐶) | |
2 | 1 | eleq2i 2680 | . 2 ⊢ (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ (𝐵 ∩ 𝐶)) |
3 | elin 3758 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
4 | 2, 3 | bitri 263 | 1 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 |
This theorem is referenced by: elin3 3766 elpredim 5609 elpred 5610 elpredg 5611 fnres 5921 funfvima 6396 fnwelem 7179 ressuppssdif 7203 fz1isolem 13102 isabl 18020 isfld 18579 2idlcpbl 19055 qus1 19056 qusrhm 19058 isidom 19125 lmres 20914 isnvc 22309 cvslvec 22733 cvsclm 22734 iscvs 22735 ishl 22966 ply1pid 23743 rplogsum 25016 isphg 27056 ishlo 27127 hhsscms 27520 mayete3i 27971 isogrp 29033 isofld 29133 sltres 31061 nofulllem5 31105 caures 32726 iscrngo 32965 fldcrng 32973 isdmn 33023 isolat 33517 srhmsubclem1 41865 srhmsubc 41868 srhmsubcALTVlem1 41884 srhmsubcALTV 41887 |
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