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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 4985. (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 4985 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | mp2an 689 | 1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 Vcvv 3473 ∩ cin 3947 {cpr 4630 ∩ cint 4950 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-un 3953 df-in 3955 df-sn 4629 df-pr 4631 df-int 4951 |
This theorem is referenced by: intprgOLD 4988 uniintsn 4991 op1stb 5471 fiint 9330 shincli 31048 chincli 31146 |
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