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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 5010 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
2 | 1 | necon1bbii 3018 | . . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅) |
3 | inteq 4668 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
4 | int0 4679 | . . . 4 ⊢ ∩ ∅ = V | |
5 | 3, 4 | syl6eq 2847 | . . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
6 | 2, 5 | sylbi 209 | . 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V) |
7 | vprc 4990 | . . 3 ⊢ ¬ V ∈ V | |
8 | eleq1 2864 | . . 3 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V)) | |
9 | 7, 8 | mtbiri 319 | . 2 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) |
10 | 6, 9 | impbii 201 | 1 ⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 = wceq 1653 ∈ wcel 2157 Vcvv 3383 ∅c0 4113 ∩ cint 4665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-v 3385 df-dif 3770 df-in 3774 df-ss 3781 df-nul 4114 df-int 4666 |
This theorem is referenced by: intabs 5015 relintabex 38658 aiotavb 41927 |
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