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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 4988 | . 2 ⊢ (𝐴 ∈ V → ∩ {𝐴} = 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∩ {𝐴} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 {csn 4627 ∩ cint 4949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-v 3476 df-un 3952 df-in 3954 df-sn 4628 df-pr 4630 df-int 4950 |
This theorem is referenced by: uniintsn 4990 intunsn 4992 op1stb 5470 op2ndb 6223 ssfii 9410 cf0 10242 cflecard 10244 uffix 23416 iotain 43161 |
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