![]() |
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 5319. (Contributed by Peter Mazsa, 19-Dec-2018.) |
Ref | Expression |
---|---|
inex2g | ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4201 | . 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
2 | inex1g 5319 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
3 | 1, 2 | eqeltrid 2833 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3471 ∩ cin 3946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-in 3954 |
This theorem is referenced by: satefvfmla1 35035 inex3 37810 inxpex 37811 dfcnvrefrels2 38000 dfcnvrefrels3 38001 iunrelexp0 43132 |
Copyright terms: Public domain | W3C validator |