Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-d0clsepcl | Unicode version |
Description: Δ0-classical logic and separation implies classical logic. (Contributed by BJ, 2-Jan-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-d0clsepcl | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4025 | . . . . . . 7 | |
2 | 1 | bj-snex 13038 | . . . . . 6 |
3 | 2 | zfauscl 4018 | . . . . 5 |
4 | eleq1 2180 | . . . . . . 7 | |
5 | eleq1 2180 | . . . . . . . 8 | |
6 | 5 | anbi1d 460 | . . . . . . 7 |
7 | 4, 6 | bibi12d 234 | . . . . . 6 |
8 | 1, 7 | spcv 2753 | . . . . 5 |
9 | 3, 8 | eximii 1566 | . . . 4 |
10 | 1 | snid 3526 | . . . . . . . 8 |
11 | 10 | biantrur 301 | . . . . . . 7 |
12 | 11 | bicomi 131 | . . . . . 6 |
13 | 12 | bibi2i 226 | . . . . 5 |
14 | 13 | exbii 1569 | . . . 4 |
15 | 9, 14 | mpbi 144 | . . 3 |
16 | bj-bd0el 12993 | . . . . 5 BOUNDED | |
17 | 16 | ax-bj-d0cl 13049 | . . . 4 DECID |
18 | dcbiit 809 | . . . 4 DECID DECID | |
19 | 17, 18 | mpbii 147 | . . 3 DECID |
20 | 15, 19 | eximii 1566 | . 2 DECID |
21 | bj-ex 12896 | . 2 DECID DECID | |
22 | 20, 21 | ax-mp 5 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 DECID wdc 804 wal 1314 wceq 1316 wex 1453 wcel 1465 c0 3333 csn 3497 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pr 4101 ax-bd0 12938 ax-bdim 12939 ax-bdor 12941 ax-bdn 12942 ax-bdal 12943 ax-bdex 12944 ax-bdeq 12945 ax-bdsep 13009 ax-bj-d0cl 13049 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-sn 3503 df-pr 3504 df-bdc 12966 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |