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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 4895 | . 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅) | |
2 | 0ss 4331 | . . 3 ⊢ ∅ ⊆ ∩ 𝐴 | |
3 | 2 | a1i 11 | . 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴) |
4 | 1, 3 | eqssd 3938 | 1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 ∅c0 4257 ∩ cint 4880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3432 df-dif 3890 df-in 3894 df-ss 3904 df-nul 4258 df-int 4881 |
This theorem is referenced by: intv 5285 inton 6317 onint0 7632 oev2 8341 ipolub00 46235 |
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