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| Description: If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.) | 
| Ref | Expression | 
|---|---|
| intnex | ⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | intex 5343 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
| 2 | 1 | necon1bbii 2989 | . . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅) | 
| 3 | inteq 4948 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
| 4 | int0 4961 | . . . 4 ⊢ ∩ ∅ = V | |
| 5 | 3, 4 | eqtrdi 2792 | . . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) | 
| 6 | 2, 5 | sylbi 217 | . 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V) | 
| 7 | vprc 5314 | . . 3 ⊢ ¬ V ∈ V | |
| 8 | eleq1 2828 | . . 3 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V)) | |
| 9 | 7, 8 | mtbiri 327 | . 2 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) | 
| 10 | 6, 9 | impbii 209 | 1 ⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∅c0 4332 ∩ cint 4945 | 
| 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-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-in 3957 df-ss 3967 df-nul 4333 df-int 4946 | 
| This theorem is referenced by: intabs 5348 relintabex 43599 aiotavb 47107 | 
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