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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 | . 2 ⊢ 𝐴 ∈ 𝐵 | |
2 | elini.2 | . 2 ⊢ 𝐴 ∈ 𝐶 | |
3 | elin 4166 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
4 | 1, 2, 3 | mpbir2an 707 | 1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∩ cin 3932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 |
This theorem is referenced by: isfin1-3 9796 isdrs2 17537 fpwipodrs 17762 0cmp 21930 comppfsc 22068 ptcmpfi 22349 alexsubALTlem2 22584 alexsubALTlem4 22586 ptcmp 22594 cnstrcvs 23672 cncvs 23676 recvs 23677 qcvs 23678 cnncvs 23690 ovolicc1 24044 ioorf 24101 corclrcl 39930 0pwfi 41198 sge0rnn0 42527 sge0reuz 42606 |
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