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Mirrors > Home > NFE Home > Th. List > ineqri | GIF version |
Description: Inference from membership to intersection. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ineqri.1 | ⊢ ((x ∈ A ∧ x ∈ B) ↔ x ∈ C) |
Ref | Expression |
---|---|
ineqri | ⊢ (A ∩ B) = C |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3219 | . . 3 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B)) | |
2 | ineqri.1 | . . 3 ⊢ ((x ∈ A ∧ x ∈ B) ↔ x ∈ C) | |
3 | 1, 2 | bitri 240 | . 2 ⊢ (x ∈ (A ∩ B) ↔ x ∈ C) |
4 | 3 | eqriv 2350 | 1 ⊢ (A ∩ B) = C |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∩ cin 3208 |
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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 |
This theorem is referenced by: inidm 3464 inass 3465 dfin2 3491 indi 3501 inab 3522 in0 3576 dmres 4986 |
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