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Mirrors > Home > NFE Home > Th. List > intunsn | GIF version |
Description: Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.) |
Ref | Expression |
---|---|
intunsn.1 | ⊢ B ∈ V |
Ref | Expression |
---|---|
intunsn | ⊢ ∩(A ∪ {B}) = (∩A ∩ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intun 3958 | . 2 ⊢ ∩(A ∪ {B}) = (∩A ∩ ∩{B}) | |
2 | intunsn.1 | . . . 4 ⊢ B ∈ V | |
3 | 2 | intsn 3962 | . . 3 ⊢ ∩{B} = B |
4 | 3 | ineq2i 3454 | . 2 ⊢ (∩A ∩ ∩{B}) = (∩A ∩ B) |
5 | 1, 4 | eqtri 2373 | 1 ⊢ ∩(A ∪ {B}) = (∩A ∩ B) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 Vcvv 2859 ∪ cun 3207 ∩ cin 3208 {csn 3737 ∩cint 3926 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-sn 3741 df-pr 3742 df-int 3927 |
This theorem is referenced by: (None) |
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