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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 5350 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
2 | 1 | necon1bbii 2988 | . . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅) |
3 | inteq 4954 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
4 | int0 4967 | . . . 4 ⊢ ∩ ∅ = V | |
5 | 3, 4 | eqtrdi 2791 | . . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
6 | 2, 5 | sylbi 217 | . 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V) |
7 | vprc 5321 | . . 3 ⊢ ¬ V ∈ V | |
8 | eleq1 2827 | . . 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 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ∩ cint 4951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-in 3970 df-ss 3980 df-nul 4340 df-int 4952 |
This theorem is referenced by: intabs 5355 relintabex 43571 aiotavb 47040 |
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