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Mirrors > Home > MPE Home > Th. List > int0el | Structured version Visualization version GIF version |
Description: The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.) |
Ref | Expression |
---|---|
int0el | ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intss1 4894 | . 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅) | |
2 | 0ss 4353 | . . 3 ⊢ ∅ ⊆ ∩ 𝐴 | |
3 | 2 | a1i 11 | . 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴) |
4 | 1, 3 | eqssd 3987 | 1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ∅c0 4294 ∩ cint 4879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-dif 3942 df-in 3946 df-ss 3955 df-nul 4295 df-int 4880 |
This theorem is referenced by: intv 5267 inton 6251 onint0 7514 oev2 8151 |
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