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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 4820 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V) | |
2 | 1 | necon1bbii 2843 | . . 3 ⊢ (¬ ∩ 𝐴 ∈ V ↔ 𝐴 = ∅) |
3 | inteq 4478 | . . . 4 ⊢ (𝐴 = ∅ → ∩ 𝐴 = ∩ ∅) | |
4 | int0 4490 | . . . 4 ⊢ ∩ ∅ = V | |
5 | 3, 4 | syl6eq 2672 | . . 3 ⊢ (𝐴 = ∅ → ∩ 𝐴 = V) |
6 | 2, 5 | sylbi 207 | . 2 ⊢ (¬ ∩ 𝐴 ∈ V → ∩ 𝐴 = V) |
7 | vprc 4796 | . . 3 ⊢ ¬ V ∈ V | |
8 | eleq1 2689 | . . 3 ⊢ (∩ 𝐴 = V → (∩ 𝐴 ∈ V ↔ V ∈ V)) | |
9 | 7, 8 | mtbiri 317 | . 2 ⊢ (∩ 𝐴 = V → ¬ ∩ 𝐴 ∈ V) |
10 | 6, 9 | impbii 199 | 1 ⊢ (¬ ∩ 𝐴 ∈ V ↔ ∩ 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ∩ cint 4475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-int 4476 |
This theorem is referenced by: intabs 4825 relintabex 37887 |
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