![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elintg | Structured version Visualization version GIF version |
Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.) (Proof shortened by JJ, 26-Jul-2021.) |
Ref | Expression |
---|---|
elintg | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2821 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
2 | 1 | ralbidv 3177 | . 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)) |
3 | dfint2 4952 | . 2 ⊢ ∩ 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} | |
4 | 2, 3 | elab2g 3670 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∩ cint 4950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-int 4951 |
This theorem is referenced by: elinti 4959 elintabg 4961 elrint 4995 onmindif 6456 onmindif2 7797 mremre 17552 toponmre 22817 1stcfb 23169 uffixfr 23647 plycpn 26026 insiga 33421 dfon2lem8 35054 trintALTVD 43943 trintALT 43944 elintd 44065 intsaluni 45344 intsal 45345 |
Copyright terms: Public domain | W3C validator |