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Mirrors > Home > MPE Home > Th. List > intnex | Structured version Visualization version GIF version |
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 4747 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
2 | 1 | necon1bbii 2831 | . . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅) |
3 | inteq 4413 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
4 | int0 4425 | . . . 4 ⊢ ∩ ∅ = V | |
5 | 3, 4 | syl6eq 2660 | . . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
6 | 2, 5 | sylbi 206 | . 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V) |
7 | vprc 4724 | . . 3 ⊢ ¬ V ∈ V | |
8 | eleq1 2676 | . . 3 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V)) | |
9 | 7, 8 | mtbiri 316 | . 2 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) |
10 | 6, 9 | impbii 198 | 1 ⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-int 4411 |
This theorem is referenced by: intabs 4752 relintabex 36906 |
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