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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 4963 | . 2 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 ⊆ ∅) | |
| 2 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ ∩ 𝐴 | |
| 3 | 2 | a1i 11 | . 2 ⊢ (∅ ∈ 𝐴 → ∅ ⊆ ∩ 𝐴) |
| 4 | 1, 3 | eqssd 4001 | 1 ⊢ (∅ ∈ 𝐴 → ∩ 𝐴 = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∅c0 4333 ∩ cint 4946 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-int 4947 |
| This theorem is referenced by: intv 5364 inton 6442 onint0 7811 oev2 8561 cuteq0 27877 ipolub00 48882 |
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