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Mirrors > Home > ILE Home > Th. List > in0 | GIF version |
Description: The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
in0 | ⊢ (𝐴 ∩ ∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3228 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | bianfi 854 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ∅)) |
3 | 2 | bicomi 123 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ∅) ↔ 𝑥 ∈ ∅) |
4 | 3 | ineqri 3130 | 1 ⊢ (𝐴 ∩ ∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∩ cin 2916 ∅c0 3224 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-in 2924 df-nul 3225 |
This theorem is referenced by: res0 4616 |
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