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Mirrors > Home > ILE Home > Th. List > 0fv | GIF version |
Description: Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.) |
Ref | Expression |
---|---|
0fv | ⊢ (∅‘𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fv 4930 | . 2 ⊢ (∅‘𝐴) = (℩𝑥𝐴∅𝑥) | |
2 | noel 3255 | . . . . . 6 ⊢ ¬ 〈𝐴, 𝑥〉 ∈ ∅ | |
3 | df-br 3786 | . . . . . 6 ⊢ (𝐴∅𝑥 ↔ 〈𝐴, 𝑥〉 ∈ ∅) | |
4 | 2, 3 | mtbir 628 | . . . . 5 ⊢ ¬ 𝐴∅𝑥 |
5 | 4 | nex 1429 | . . . 4 ⊢ ¬ ∃𝑥 𝐴∅𝑥 |
6 | euex 1971 | . . . 4 ⊢ (∃!𝑥 𝐴∅𝑥 → ∃𝑥 𝐴∅𝑥) | |
7 | 5, 6 | mto 620 | . . 3 ⊢ ¬ ∃!𝑥 𝐴∅𝑥 |
8 | iotanul 4902 | . . 3 ⊢ (¬ ∃!𝑥 𝐴∅𝑥 → (℩𝑥𝐴∅𝑥) = ∅) | |
9 | 7, 8 | ax-mp 7 | . 2 ⊢ (℩𝑥𝐴∅𝑥) = ∅ |
10 | 1, 9 | eqtri 2101 | 1 ⊢ (∅‘𝐴) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∃!weu 1941 ∅c0 3251 〈cop 3401 class class class wbr 3785 ℩cio 4885 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-in 2979 df-ss 2986 df-nul 3252 df-sn 3404 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 |
This theorem is referenced by: (None) |
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