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Mirrors > Home > MPE Home > Th. List > intpr | Structured version Visualization version GIF version |
Description: The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.) Prove from intprg 4984. (Revised by BJ, 1-Sep-2024.) |
Ref | Expression |
---|---|
intpr.1 | ⊢ 𝐴 ∈ V |
intpr.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
intpr | ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intpr.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | intpr.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | intprg 4984 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∩ cin 3946 {cpr 4629 ∩ 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-v 3476 df-un 3952 df-in 3954 df-sn 4628 df-pr 4630 df-int 4950 |
This theorem is referenced by: intprgOLD 4987 uniintsn 4990 op1stb 5470 fiint 9320 shincli 30602 chincli 30700 |
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