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Mirrors > Home > ILE Home > Th. List > intprg | GIF version |
Description: The intersection of a pair is the intersection of its members. Closed form of intpr 3773. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.) |
Ref | Expression |
---|---|
intprg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 3570 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦}) | |
2 | 1 | inteqd 3746 | . . 3 ⊢ (𝑥 = 𝐴 → ∩ {𝑥, 𝑦} = ∩ {𝐴, 𝑦}) |
3 | ineq1 3240 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦)) | |
4 | 2, 3 | eqeq12d 2132 | . 2 ⊢ (𝑥 = 𝐴 → (∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦) ↔ ∩ {𝐴, 𝑦} = (𝐴 ∩ 𝑦))) |
5 | preq2 3571 | . . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵}) | |
6 | 5 | inteqd 3746 | . . 3 ⊢ (𝑦 = 𝐵 → ∩ {𝐴, 𝑦} = ∩ {𝐴, 𝐵}) |
7 | ineq2 3241 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵)) | |
8 | 6, 7 | eqeq12d 2132 | . 2 ⊢ (𝑦 = 𝐵 → (∩ {𝐴, 𝑦} = (𝐴 ∩ 𝑦) ↔ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))) |
9 | vex 2663 | . . 3 ⊢ 𝑥 ∈ V | |
10 | vex 2663 | . . 3 ⊢ 𝑦 ∈ V | |
11 | 9, 10 | intpr 3773 | . 2 ⊢ ∩ {𝑥, 𝑦} = (𝑥 ∩ 𝑦) |
12 | 4, 8, 11 | vtocl2g 2724 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ∩ cin 3040 {cpr 3498 ∩ cint 3741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-v 2662 df-un 3045 df-in 3047 df-sn 3503 df-pr 3504 df-int 3742 |
This theorem is referenced by: intsng 3775 op1stbg 4370 |
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