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Mirrors > Home > ILE Home > Th. List > eliin | GIF version |
Description: Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.) |
Ref | Expression |
---|---|
eliin | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2251 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
2 | 1 | ralbidv 2489 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
3 | df-iin 3903 | . 2 ⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶} | |
4 | 2, 3 | elab2g 2898 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1363 ∈ wcel 2159 ∀wral 2467 ∩ ciin 3901 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2170 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ral 2472 df-v 2753 df-iin 3903 |
This theorem is referenced by: iinconstm 3909 iuniin 3910 iinss1 3912 ssiinf 3950 iinss 3952 iinss2 3953 iinab 3962 iundif2ss 3966 iindif2m 3968 iinin2m 3969 elriin 3971 iinpw 3991 xpiindim 4778 cnviinm 5184 iinerm 6624 ixpiinm 6741 |
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