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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > inex3 | Structured version Visualization version GIF version |
Description: Sufficient condition for the intersection relation to be a set. (Contributed by Peter Mazsa, 24-Nov-2019.) |
Ref | Expression |
---|---|
inex3 | ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inex1g 5114 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
2 | inex2g 5115 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∩ 𝐵) ∈ V) | |
3 | 1, 2 | jaoi 852 | 1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 842 ∈ wcel 2081 Vcvv 3437 ∩ cin 3858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5094 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-rab 3114 df-v 3439 df-in 3866 |
This theorem is referenced by: (None) |
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