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Mirrors > Home > ILE Home > Th. List > abvor0dc | Unicode version |
Description: The class builder of a decidable proposition not containing the abstraction variable is either the universal class or the empty set. (Contributed by Jim Kingdon, 1-Aug-2018.) |
Ref | Expression |
---|---|
abvor0dc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 743 | . 2 DECID | |
2 | id 19 | . . . . 5 | |
3 | vex 2560 | . . . . . 6 | |
4 | 3 | a1i 9 | . . . . 5 |
5 | 2, 4 | 2thd 164 | . . . 4 |
6 | 5 | abbi1dv 2157 | . . 3 |
7 | id 19 | . . . . 5 | |
8 | noel 3228 | . . . . . 6 | |
9 | 8 | a1i 9 | . . . . 5 |
10 | 7, 9 | 2falsed 618 | . . . 4 |
11 | 10 | abbi1dv 2157 | . . 3 |
12 | 6, 11 | orim12i 676 | . 2 |
13 | 1, 12 | sylbi 114 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wo 629 DECID wdc 742 wceq 1243 wcel 1393 cab 2026 cvv 2557 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-dc 743 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-nul 3225 |
This theorem is referenced by: (None) |
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