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Mirrors > Home > NFE 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 625 | . 2 ⊢ ((x ∈ A ∧ x ∈ A) ↔ x ∈ A) | |
2 | 1 | ineqri 3449 | 1 ⊢ (A ∩ A) = A |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 ∩ cin 3208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 |
This theorem is referenced by: inindi 3472 inindir 3473 uneqin 3506 ssdifeq0 3632 intsng 3961 xpindi 4864 xpindir 4865 xpassen 6057 |
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