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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 4892. (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 4892 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∩ cin 3865 {cpr 4543 ∩ cint 4859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-un 3871 df-in 3873 df-sn 4542 df-pr 4544 df-int 4860 |
This theorem is referenced by: intprgOLD 4895 uniintsn 4898 op1stb 5355 fiint 8948 shincli 29443 chincli 29541 |
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