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Mirrors > Home > MPE Home > Th. List > ineqri | Structured version Visualization version GIF version |
Description: Inference from membership to intersection. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
ineqri.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) |
Ref | Expression |
---|---|
ineqri | ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 4090 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | ineqri.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) | |
3 | 1, 2 | bitri 276 | . 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥 ∈ 𝐶) |
4 | 3 | eqriv 2792 | 1 ⊢ (𝐴 ∩ 𝐵) = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∩ cin 3858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-v 3439 df-in 3866 |
This theorem is referenced by: inidm 4115 inass 4116 dfin2 4157 indi 4170 inab 4191 in0 4265 pwin 5342 dfres3 5739 dmres 5756 inixp 34535 |
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