| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 4981. (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 4981 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 {cpr 4628 ∩ cint 4946 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-in 3958 df-sn 4627 df-pr 4629 df-int 4947 |
| This theorem is referenced by: uniintsn 4985 op1stb 5476 fiint 9366 fiintOLD 9367 shincli 31381 chincli 31479 |
| Copyright terms: Public domain | W3C validator |