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Mirrors > Home > MPE Home > Th. List > intsn | Structured version Visualization version GIF version |
Description: The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.) |
Ref | Expression |
---|---|
intsn.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
intsn | ⊢ ∩ {𝐴} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | intsng 4911 | . 2 ⊢ (𝐴 ∈ V → ∩ {𝐴} = 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∩ {𝐴} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 ∩ cint 4876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-un 3941 df-in 3943 df-sn 4568 df-pr 4570 df-int 4877 |
This theorem is referenced by: uniintsn 4913 intunsn 4915 op1stb 5363 op2ndb 6084 ssfii 8883 cf0 9673 cflecard 9675 uffix 22529 iotain 40769 |
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