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Mirrors > Home > ILE Home > Th. List > intpr | 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.) |
Ref | Expression |
---|---|
intpr.1 | ⊢ 𝐴 ∈ V |
intpr.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
intpr | ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.26 1410 | . . . 4 ⊢ (∀𝑦((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)) ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) | |
2 | vex 2604 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
3 | 2 | elpr 3419 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) |
4 | 3 | imbi1i 236 | . . . . . 6 ⊢ ((𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦)) |
5 | jaob 663 | . . . . . 6 ⊢ (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) | |
6 | 4, 5 | bitri 182 | . . . . 5 ⊢ ((𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) |
7 | 6 | albii 1399 | . . . 4 ⊢ (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ∀𝑦((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) |
8 | intpr.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
9 | 8 | clel4 2731 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦)) |
10 | intpr.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
11 | 10 | clel4 2731 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦)) |
12 | 9, 11 | anbi12i 447 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) |
13 | 1, 7, 12 | 3bitr4i 210 | . . 3 ⊢ (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
14 | vex 2604 | . . . 4 ⊢ 𝑥 ∈ V | |
15 | 14 | elint 3642 | . . 3 ⊢ (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦)) |
16 | elin 3155 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
17 | 13, 15, 16 | 3bitr4i 210 | . 2 ⊢ (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴 ∩ 𝐵)) |
18 | 17 | eqriv 2078 | 1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∨ wo 661 ∀wal 1282 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ∩ cin 2972 {cpr 3399 ∩ cint 3636 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-sn 3404 df-pr 3405 df-int 3637 |
This theorem is referenced by: intprg 3669 op1stb 4227 onintexmid 4315 |
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