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Mirrors > Home > ILE Home > Th. List > elinti | GIF version |
Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) |
Ref | Expression |
---|---|
elinti | ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elintg 3867 | . . 3 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)) | |
2 | eleq2 2253 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐶)) | |
3 | 2 | rspccv 2853 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)) |
4 | 1, 3 | biimtrdi 163 | . 2 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶))) |
5 | 4 | pm2.43i 49 | 1 ⊢ (𝐴 ∈ ∩ 𝐵 → (𝐶 ∈ 𝐵 → 𝐴 ∈ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ∀wral 2468 ∩ cint 3859 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-int 3860 |
This theorem is referenced by: subgintm 13154 subrngintm 13576 subrgintm 13607 lssintclm 13717 |
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