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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 5190 | . 2 ⊢ (∅‘𝐴) = (℩𝑥𝐴∅𝑥) | |
2 | noel 3408 | . . . . . 6 ⊢ ¬ 〈𝐴, 𝑥〉 ∈ ∅ | |
3 | df-br 3977 | . . . . . 6 ⊢ (𝐴∅𝑥 ↔ 〈𝐴, 𝑥〉 ∈ ∅) | |
4 | 2, 3 | mtbir 661 | . . . . 5 ⊢ ¬ 𝐴∅𝑥 |
5 | 4 | nex 1487 | . . . 4 ⊢ ¬ ∃𝑥 𝐴∅𝑥 |
6 | euex 2043 | . . . 4 ⊢ (∃!𝑥 𝐴∅𝑥 → ∃𝑥 𝐴∅𝑥) | |
7 | 5, 6 | mto 652 | . . 3 ⊢ ¬ ∃!𝑥 𝐴∅𝑥 |
8 | iotanul 5162 | . . 3 ⊢ (¬ ∃!𝑥 𝐴∅𝑥 → (℩𝑥𝐴∅𝑥) = ∅) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ (℩𝑥𝐴∅𝑥) = ∅ |
10 | 1, 9 | eqtri 2185 | 1 ⊢ (∅‘𝐴) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1342 ∃wex 1479 ∃!weu 2013 ∈ wcel 2135 ∅c0 3404 〈cop 3573 class class class wbr 3976 ℩cio 5145 ‘cfv 5182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-in 3117 df-ss 3124 df-nul 3405 df-sn 3576 df-uni 3784 df-br 3977 df-iota 5147 df-fv 5190 |
This theorem is referenced by: strsl0 12385 |
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