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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 393 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴) | |
2 | 1 | ineqri 3269 | 1 ⊢ (𝐴 ∩ 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 ∩ cin 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-in 3077 |
This theorem is referenced by: inindi 3293 inindir 3294 uneqin 3327 ssdifeq0 3445 intsng 3805 xpindi 4674 xpindir 4675 resindm 4861 ofres 5996 offval2 5997 ofrfval2 5998 suppssof1 5999 ofco 6000 offveqb 6001 caofref 6003 caofrss 6006 caoftrn 6007 undifdc 6812 baspartn 12217 epttop 12259 dvaddxxbr 12834 dvmulxxbr 12835 dvaddxx 12836 dvmulxx 12837 dviaddf 12838 dvimulf 12839 |
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