|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > inex2g | Structured version Visualization version GIF version | ||
| Description: Sufficient condition for an intersection to be a set. Commuted form of inex1g 5318. (Contributed by Peter Mazsa, 19-Dec-2018.) | 
| Ref | Expression | 
|---|---|
| inex2g | ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | incom 4208 | . 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
| 2 | inex1g 5318 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
| 3 | 1, 2 | eqeltrid 2844 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-in 3957 | 
| This theorem is referenced by: satefvfmla1 35431 inex3 38340 inxpex 38341 dfcnvrefrels2 38530 dfcnvrefrels3 38531 iunrelexp0 43720 | 
| Copyright terms: Public domain | W3C validator |