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Mirrors > Home > ILE Home > Th. List > elind | GIF version |
Description: Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
elind.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
elind.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
elind | ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elind.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
2 | elind.2 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | elin 3318 | . 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) | |
4 | 1, 2, 3 | sylanbrc 417 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ∩ cin 3128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-in 3135 |
This theorem is referenced by: elfir 6971 infpwfidom 7196 nninfdclemcl 12448 nninfdclemp1 12450 strslfv2d 12504 insubm 12871 baspartn 13486 bastg 13497 isopn3 13561 restbasg 13604 lmss 13682 metrest 13942 tgioo 13982 dvmulxxbr 14102 pilem3 14140 2sqlem7 14404 |
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