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Mirrors > Home > ILE Home > Th. List > inidm | GIF version |
Description: Idempotent law for intersection of classes. Theorem 15 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
inidm | ⊢ (𝐴 ∩ 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anidm 396 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴) | |
2 | 1 | ineqri 3328 | 1 ⊢ (𝐴 ∩ 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 ∩ cin 3128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 |
This theorem is referenced by: inindi 3352 inindir 3353 uneqin 3386 ssdifeq0 3505 intsng 3876 xpindi 4758 xpindir 4759 resindm 4945 ofres 6091 offval2 6092 ofrfval2 6093 suppssof1 6094 ofco 6095 offveqb 6096 caofref 6098 caofrss 6101 caoftrn 6102 undifdc 6917 strressid 12509 ressinbasd 12512 baspartn 13208 epttop 13250 dvaddxxbr 13825 dvmulxxbr 13826 dvaddxx 13827 dvmulxx 13828 dviaddf 13829 dvimulf 13830 |
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