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Mirrors > Home > MPE Home > Th. List > intsng | Structured version Visualization version 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 4330 | . . 3 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | inteqi 4616 | . 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴} |
3 | intprg 4646 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) | |
4 | 3 | anidms 556 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴)) |
5 | inidm 3971 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
6 | 4, 5 | syl6eq 2821 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴) |
7 | 2, 6 | syl5eq 2817 | 1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ∩ cin 3722 {csn 4317 {cpr 4319 ∩ cint 4612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-v 3353 df-un 3728 df-in 3730 df-sn 4318 df-pr 4320 df-int 4613 |
This theorem is referenced by: intsn 4648 riinint 5519 bj-snmoore 33399 elrfi 37783 |
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