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Mirrors > Home > NFE Home > Th. List > iinconst | GIF version |
Description: Indexed intersection of a constant class, i.e. where B does not depend on x. (Contributed by Mario Carneiro, 6-Feb-2015.) |
Ref | Expression |
---|---|
iinconst | ⊢ (A ≠ ∅ → ∩x ∈ A B = B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.3rzv 3643 | . . 3 ⊢ (A ≠ ∅ → (y ∈ B ↔ ∀x ∈ A y ∈ B)) | |
2 | vex 2862 | . . . 4 ⊢ y ∈ V | |
3 | eliin 3974 | . . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B)) | |
4 | 2, 3 | ax-mp 8 | . . 3 ⊢ (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B) |
5 | 1, 4 | syl6rbbr 255 | . 2 ⊢ (A ≠ ∅ → (y ∈ ∩x ∈ A B ↔ y ∈ B)) |
6 | 5 | eqrdv 2351 | 1 ⊢ (A ≠ ∅ → ∩x ∈ A B = B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∀wral 2614 Vcvv 2859 ∅c0 3550 ∩ciin 3970 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-ne 2518 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-nul 3551 df-iin 3972 |
This theorem is referenced by: (None) |
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