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Mirrors > Home > NFE Home > Th. List > int0el | 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 3941 | . 2 ⊢ (∅ ∈ A → ∩A ⊆ ∅) | |
2 | 0ss 3579 | . . 3 ⊢ ∅ ⊆ ∩A | |
3 | 2 | a1i 10 | . 2 ⊢ (∅ ∈ A → ∅ ⊆ ∩A) |
4 | 1, 3 | eqssd 3289 | 1 ⊢ (∅ ∈ A → ∩A = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 ⊆ wss 3257 ∅c0 3550 ∩cint 3926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 df-int 3927 |
This theorem is referenced by: (None) |
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