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Mirrors > Home > ILE Home > Th. List > 0in | GIF version |
Description: The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
0in | ⊢ (∅ ∩ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 3342 | . 2 ⊢ (∅ ∩ 𝐴) = (𝐴 ∩ ∅) | |
2 | in0 3472 | . 2 ⊢ (𝐴 ∩ ∅) = ∅ | |
3 | 1, 2 | eqtri 2210 | 1 ⊢ (∅ ∩ 𝐴) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∩ cin 3143 ∅c0 3437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-in 3150 df-nul 3438 |
This theorem is referenced by: setsfun 12546 setsfun0 12547 restsn 14132 |
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