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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 4910 | . 2 ⊢ (∅‘𝐴) = (℩𝑥𝐴∅𝑥) | |
2 | noel 3228 | . . . . . 6 ⊢ ¬ 〈𝐴, 𝑥〉 ∈ ∅ | |
3 | df-br 3765 | . . . . . 6 ⊢ (𝐴∅𝑥 ↔ 〈𝐴, 𝑥〉 ∈ ∅) | |
4 | 2, 3 | mtbir 596 | . . . . 5 ⊢ ¬ 𝐴∅𝑥 |
5 | 4 | nex 1389 | . . . 4 ⊢ ¬ ∃𝑥 𝐴∅𝑥 |
6 | euex 1930 | . . . 4 ⊢ (∃!𝑥 𝐴∅𝑥 → ∃𝑥 𝐴∅𝑥) | |
7 | 5, 6 | mto 588 | . . 3 ⊢ ¬ ∃!𝑥 𝐴∅𝑥 |
8 | iotanul 4882 | . . 3 ⊢ (¬ ∃!𝑥 𝐴∅𝑥 → (℩𝑥𝐴∅𝑥) = ∅) | |
9 | 7, 8 | ax-mp 7 | . 2 ⊢ (℩𝑥𝐴∅𝑥) = ∅ |
10 | 1, 9 | eqtri 2060 | 1 ⊢ (∅‘𝐴) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∃!weu 1900 ∅c0 3224 〈cop 3378 class class class wbr 3764 ℩cio 4865 ‘cfv 4902 |
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-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 |
This theorem is referenced by: (None) |
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