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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 2872 | . 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ 𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑})) | |
2 | eleq1 2872 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
3 | 2 | imbi2d 342 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑦 ∈ 𝑥) ↔ (𝜑 → 𝐴 ∈ 𝑥))) |
4 | 3 | ralbidv 3166 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 (𝜑 → 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))) |
5 | vex 3443 | . . 3 ⊢ 𝑦 ∈ V | |
6 | 5 | elintrab 4800 | . 2 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝑦 ∈ 𝑥)) |
7 | 1, 4, 6 | vtoclbg 3514 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐵 (𝜑 → 𝐴 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1525 ∈ wcel 2083 ∀wral 3107 {crab 3111 ∩ cint 4788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rab 3116 df-v 3442 df-int 4789 |
This theorem is referenced by: tskmid 10115 eltskm 10118 ldsysgenld 31032 ldgenpisyslem1 31035 elpcliN 36581 elmapintrab 39442 |
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