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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 5280. (Contributed by Peter Mazsa, 19-Dec-2018.) |
Ref | Expression |
---|---|
inex2g | ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4165 | . 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
2 | inex1g 5280 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
3 | 1, 2 | eqeltrid 2838 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3447 ∩ cin 3913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-in 3921 |
This theorem is referenced by: satefvfmla1 34083 inex3 36849 inxpex 36850 dfcnvrefrels2 37040 dfcnvrefrels3 37041 iunrelexp0 42066 |
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