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Mirrors > Home > NFE Home > Th. List > intsng | GIF version |
Description: Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
intsng | ⊢ (A ∈ V → ∩{A} = A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 3747 | . . 3 ⊢ {A} = {A, A} | |
2 | 1 | inteqi 3930 | . 2 ⊢ ∩{A} = ∩{A, A} |
3 | intprg 3960 | . . . 4 ⊢ ((A ∈ V ∧ A ∈ V) → ∩{A, A} = (A ∩ A)) | |
4 | 3 | anidms 626 | . . 3 ⊢ (A ∈ V → ∩{A, A} = (A ∩ A)) |
5 | inidm 3464 | . . 3 ⊢ (A ∩ A) = A | |
6 | 4, 5 | syl6eq 2401 | . 2 ⊢ (A ∈ V → ∩{A, A} = A) |
7 | 2, 6 | syl5eq 2397 | 1 ⊢ (A ∈ V → ∩{A} = A) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 ∩ cin 3208 {csn 3737 {cpr 3738 ∩cint 3926 |
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-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: intsn 3962 |
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