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Mirrors > Home > ILE Home > Th. List > elintrab | GIF version |
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.) |
Ref | Expression |
---|---|
inteqab.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
elintrab | ⊢ (A ∈ ∩ {x ∈ B ∣ φ} ↔ ∀x ∈ B (φ → A ∈ x)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inteqab.1 | . . . 4 ⊢ A ∈ V | |
2 | 1 | elintab 3617 | . . 3 ⊢ (A ∈ ∩ {x ∣ (x ∈ B ∧ φ)} ↔ ∀x((x ∈ B ∧ φ) → A ∈ x)) |
3 | impexp 250 | . . . 4 ⊢ (((x ∈ B ∧ φ) → A ∈ x) ↔ (x ∈ B → (φ → A ∈ x))) | |
4 | 3 | albii 1356 | . . 3 ⊢ (∀x((x ∈ B ∧ φ) → A ∈ x) ↔ ∀x(x ∈ B → (φ → A ∈ x))) |
5 | 2, 4 | bitri 173 | . 2 ⊢ (A ∈ ∩ {x ∣ (x ∈ B ∧ φ)} ↔ ∀x(x ∈ B → (φ → A ∈ x))) |
6 | df-rab 2309 | . . . 4 ⊢ {x ∈ B ∣ φ} = {x ∣ (x ∈ B ∧ φ)} | |
7 | 6 | inteqi 3610 | . . 3 ⊢ ∩ {x ∈ B ∣ φ} = ∩ {x ∣ (x ∈ B ∧ φ)} |
8 | 7 | eleq2i 2101 | . 2 ⊢ (A ∈ ∩ {x ∈ B ∣ φ} ↔ A ∈ ∩ {x ∣ (x ∈ B ∧ φ)}) |
9 | df-ral 2305 | . 2 ⊢ (∀x ∈ B (φ → A ∈ x) ↔ ∀x(x ∈ B → (φ → A ∈ x))) | |
10 | 5, 8, 9 | 3bitr4i 201 | 1 ⊢ (A ∈ ∩ {x ∈ B ∣ φ} ↔ ∀x ∈ B (φ → A ∈ x)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 ∈ wcel 1390 {cab 2023 ∀wral 2300 {crab 2304 Vcvv 2551 ∩ cint 3606 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rab 2309 df-v 2553 df-int 3607 |
This theorem is referenced by: elintrabg 3619 intmin 3626 bj-indint 9390 |
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