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Mirrors > Home > MPE Home > Th. List > elini | Structured version Visualization version GIF version |
Description: Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
elini.1 | ⊢ 𝐴 ∈ 𝐵 |
elini.2 | ⊢ 𝐴 ∈ 𝐶 |
Ref | Expression |
---|---|
elini | ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elini.1 | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
2 | elini.2 | . . 3 ⊢ 𝐴 ∈ 𝐶 | |
3 | 1, 2 | pm3.2i 470 | . 2 ⊢ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) |
4 | elin 3758 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
5 | 3, 4 | mpbir 220 | 1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 1977 ∩ cin 3539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 |
This theorem is referenced by: recvs 22754 qcvs 22755 cnncvs 22767 0pwfi 38252 sge0rnn0 39261 sge0reuz 39340 |
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