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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.) |
Ref | Expression |
---|---|
intpr.1 | ⊢ 𝐴 ∈ V |
intpr.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
intpr | ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.26 1798 | . . . 4 ⊢ (∀𝑦((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦)) ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) | |
2 | vex 3203 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
3 | 2 | elpr 4198 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐴, 𝐵} ↔ (𝑦 = 𝐴 ∨ 𝑦 = 𝐵)) |
4 | 3 | imbi1i 339 | . . . . . 6 ⊢ ((𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦)) |
5 | jaob 822 | . . . . . 6 ⊢ (((𝑦 = 𝐴 ∨ 𝑦 = 𝐵) → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) | |
6 | 4, 5 | bitri 264 | . . . . 5 ⊢ ((𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ ((𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ (𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) |
7 | 6 | albii 1747 | . . . 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 3342 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ ∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦)) |
10 | intpr.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
11 | 10 | clel4 3342 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦)) |
12 | 9, 11 | anbi12i 733 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (∀𝑦(𝑦 = 𝐴 → 𝑥 ∈ 𝑦) ∧ ∀𝑦(𝑦 = 𝐵 → 𝑥 ∈ 𝑦))) |
13 | 1, 7, 12 | 3bitr4i 292 | . . 3 ⊢ (∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
14 | vex 3203 | . . . 4 ⊢ 𝑥 ∈ V | |
15 | 14 | elint 4481 | . . 3 ⊢ (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ ∀𝑦(𝑦 ∈ {𝐴, 𝐵} → 𝑥 ∈ 𝑦)) |
16 | elin 3796 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
17 | 13, 15, 16 | 3bitr4i 292 | . 2 ⊢ (𝑥 ∈ ∩ {𝐴, 𝐵} ↔ 𝑥 ∈ (𝐴 ∩ 𝐵)) |
18 | 17 | eqriv 2619 | 1 ⊢ ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 {cpr 4179 ∩ cint 4475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-in 3581 df-sn 4178 df-pr 4180 df-int 4476 |
This theorem is referenced by: intprg 4511 uniintsn 4514 op1stb 4940 fiint 8237 shincli 28221 chincli 28319 |
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