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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 3967 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | |
| 4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∩ cin 3950 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 |
| This theorem is referenced by: isfin1-3 10426 setc2ohom 18140 isdrs2 18352 fpwipodrs 18585 0cmp 23402 comppfsc 23540 ptcmpfi 23821 alexsubALTlem2 24056 alexsubALTlem4 24058 ptcmp 24066 cnstrcvs 25174 cncvs 25178 recvs 25179 recvsOLD 25180 qcvs 25181 cnncvs 25193 ovolicc1 25551 ioorf 25608 zringpid 33580 corclrcl 43720 0pwfi 45064 sge0rnn0 46383 sge0reuz 46462 termc2 49148 |
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