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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 4884 | . 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅) | |
2 | 0ss 4350 | . . 3 ⊢ ∅ ⊆ ∩ 𝐴 | |
3 | 2 | a1i 11 | . 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴) |
4 | 1, 3 | eqssd 3984 | 1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ∅c0 4291 ∩ cint 4869 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-dif 3939 df-in 3943 df-ss 3952 df-nul 4292 df-int 4870 |
This theorem is referenced by: intv 5257 inton 6243 onint0 7505 oev2 8142 |
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