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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 5318 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
| 2 | inex2g 5319 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∩ 𝐵) ∈ V) | |
| 3 | 1, 2 | jaoi 857 | 1 ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → (𝐴 ∩ 𝐵) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∨ wo 847 ∈ 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-or 848 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: (None) | 
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