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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 4169 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ∩ cin 3935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 |
This theorem is referenced by: isfin1-3 9808 isdrs2 17549 fpwipodrs 17774 0cmp 22002 comppfsc 22140 ptcmpfi 22421 alexsubALTlem2 22656 alexsubALTlem4 22658 ptcmp 22666 cnstrcvs 23745 cncvs 23749 recvs 23750 qcvs 23751 cnncvs 23763 ovolicc1 24117 ioorf 24174 corclrcl 40072 0pwfi 41341 sge0rnn0 42670 sge0reuz 42749 |
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