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Mirrors > Home > MPE Home > Th. List > elintrabg | Structured version Visualization version GIF version |
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.) |
Ref | Expression |
---|---|
elintrabg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2900 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑})) | |
2 | eleq1 2900 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
3 | 2 | imbi2d 343 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝐴 ∈ 𝑥))) |
4 | 3 | ralbidv 3197 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 (𝜑 → 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))) |
5 | vex 3497 | . . 3 ⊢ 𝑦 ∈ V | |
6 | 5 | elintrab 4887 | . 2 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝑦 ∈ 𝑥)) |
7 | 1, 4, 6 | vtoclbg 3568 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 ∩ cint 4875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-int 4876 |
This theorem is referenced by: tskmid 10261 eltskm 10264 ldsysgenld 31419 ldgenpisyslem1 31422 elpcliN 37028 elmapintrab 39934 |
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