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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 5204 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
2 | 1 | necon1bbii 3036 | . . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅) |
3 | inteq 4841 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
4 | int0 4852 | . . . 4 ⊢ ∩ ∅ = V | |
5 | 3, 4 | eqtrdi 2849 | . . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
6 | 2, 5 | sylbi 220 | . 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V) |
7 | vprc 5183 | . . 3 ⊢ ¬ V ∈ V | |
8 | eleq1 2877 | . . 3 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V)) | |
9 | 7, 8 | mtbiri 330 | . 2 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) |
10 | 6, 9 | impbii 212 | 1 ⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 ∩ cint 4838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 ax-sep 5167 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-int 4839 |
This theorem is referenced by: intabs 5209 relintabex 40281 aiotavb 43647 |
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