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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 5216. (Contributed by Peter Mazsa, 19-Dec-2018.) |
Ref | Expression |
---|---|
inex2g | ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4171 | . 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
2 | inex1g 5216 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
3 | 1, 2 | eqeltrid 2916 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∩ 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 Vcvv 3491 ∩ cin 3928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-in 3936 |
This theorem is referenced by: satefvfmla1 32693 inex3 35628 inxpex 35629 dfcnvrefrels2 35799 dfcnvrefrels3 35800 |
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