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Mirrors > Home > NFE Home > Th. List > elriin | GIF version |
Description: Elementhood in a relative intersection. (Contributed by Mario Carneiro, 30-Dec-2016.) |
Ref | Expression |
---|---|
elriin | ⊢ (B ∈ (A ∩ ∩x ∈ X S) ↔ (B ∈ A ∧ ∀x ∈ X B ∈ S)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3219 | . 2 ⊢ (B ∈ (A ∩ ∩x ∈ X S) ↔ (B ∈ A ∧ B ∈ ∩x ∈ X S)) | |
2 | eliin 3974 | . . 3 ⊢ (B ∈ A → (B ∈ ∩x ∈ X S ↔ ∀x ∈ X B ∈ S)) | |
3 | 2 | pm5.32i 618 | . 2 ⊢ ((B ∈ A ∧ B ∈ ∩x ∈ X S) ↔ (B ∈ A ∧ ∀x ∈ X B ∈ S)) |
4 | 1, 3 | bitri 240 | 1 ⊢ (B ∈ (A ∩ ∩x ∈ X S) ↔ (B ∈ A ∧ ∀x ∈ X B ∈ S)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∈ wcel 1710 ∀wral 2614 ∩ cin 3208 ∩ciin 3970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-iin 3972 |
This theorem is referenced by: (None) |
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