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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 1481 | . . . 4 ⊢ (∀𝑦((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)) ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) | |
2 | vex 2740 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
3 | 2 | elpr 3613 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) |
4 | 3 | imbi1i 238 | . . . . . 6 ⊢ ((𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦)) |
5 | jaob 710 | . . . . . 6 ⊢ (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) | |
6 | 4, 5 | bitri 184 | . . . . 5 ⊢ ((𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) |
7 | 6 | albii 1470 | . . . 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 2873 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦)) |
10 | intpr.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
11 | 10 | clel4 2873 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦)) |
12 | 9, 11 | anbi12i 460 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) |
13 | 1, 7, 12 | 3bitr4i 212 | . . 3 ⊢ (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
14 | vex 2740 | . . . 4 ⊢ 𝑥 ∈ V | |
15 | 14 | elint 3850 | . . 3 ⊢ (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦)) |
16 | elin 3318 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
17 | 13, 15, 16 | 3bitr4i 212 | . 2 ⊢ (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴 ∩ 𝐵)) |
18 | 17 | eqriv 2174 | 1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 ∀wal 1351 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∩ cin 3128 {cpr 3593 ∩ cint 3844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-sn 3598 df-pr 3599 df-int 3845 |
This theorem is referenced by: intprg 3877 op1stb 4477 onintexmid 4571 |
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