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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 4853 | . 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅) | |
2 | 0ss 4304 | . . 3 ⊢ ∅ ⊆ ∩ 𝐴 | |
3 | 2 | a1i 11 | . 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴) |
4 | 1, 3 | eqssd 3932 | 1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 ∩ cint 4838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 df-int 4839 |
This theorem is referenced by: intv 5229 inton 6216 onint0 7491 oev2 8131 |
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