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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 4968 | . 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅) | |
2 | 0ss 4406 | . . 3 ⊢ ∅ ⊆ ∩ 𝐴 | |
3 | 2 | a1i 11 | . 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴) |
4 | 1, 3 | eqssd 4013 | 1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∅c0 4339 ∩ cint 4951 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-ss 3980 df-nul 4340 df-int 4952 |
This theorem is referenced by: intv 5370 inton 6444 onint0 7811 oev2 8560 cuteq0 27892 ipolub00 48782 |
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