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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 4853 | . 2 ⊢ (∅‘A) = (℩xA∅x) | |
2 | noel 3222 | . . . . . 6 ⊢ ¬ 〈A, x〉 ∈ ∅ | |
3 | df-br 3756 | . . . . . 6 ⊢ (A∅x ↔ 〈A, x〉 ∈ ∅) | |
4 | 2, 3 | mtbir 595 | . . . . 5 ⊢ ¬ A∅x |
5 | 4 | nex 1386 | . . . 4 ⊢ ¬ ∃x A∅x |
6 | euex 1927 | . . . 4 ⊢ (∃!x A∅x → ∃x A∅x) | |
7 | 5, 6 | mto 587 | . . 3 ⊢ ¬ ∃!x A∅x |
8 | iotanul 4825 | . . 3 ⊢ (¬ ∃!x A∅x → (℩xA∅x) = ∅) | |
9 | 7, 8 | ax-mp 7 | . 2 ⊢ (℩xA∅x) = ∅ |
10 | 1, 9 | eqtri 2057 | 1 ⊢ (∅‘A) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∃!weu 1897 ∅c0 3218 〈cop 3370 class class class wbr 3755 ℩cio 4808 ‘cfv 4845 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-in 2918 df-ss 2925 df-nul 3219 df-sn 3373 df-uni 3572 df-br 3756 df-iota 4810 df-fv 4853 |
This theorem is referenced by: (None) |
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