Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > suppssov1 | Unicode version |
Description: Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
Ref | Expression |
---|---|
suppssov1.s | |
suppssov1.o | |
suppssov1.a | |
suppssov1.b |
Ref | Expression |
---|---|
suppssov1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suppssov1.a | . . . . . . . 8 | |
2 | elex 2671 | . . . . . . . 8 | |
3 | 1, 2 | syl 14 | . . . . . . 7 |
4 | 3 | adantr 274 | . . . . . 6 |
5 | eldifsni 3622 | . . . . . . . 8 | |
6 | oveq2 5750 | . . . . . . . . . . . 12 | |
7 | 6 | eqeq1d 2126 | . . . . . . . . . . 11 |
8 | suppssov1.o | . . . . . . . . . . . . 13 | |
9 | 8 | ralrimiva 2482 | . . . . . . . . . . . 12 |
10 | 9 | adantr 274 | . . . . . . . . . . 11 |
11 | suppssov1.b | . . . . . . . . . . 11 | |
12 | 7, 10, 11 | rspcdva 2768 | . . . . . . . . . 10 |
13 | oveq1 5749 | . . . . . . . . . . 11 | |
14 | 13 | eqeq1d 2126 | . . . . . . . . . 10 |
15 | 12, 14 | syl5ibrcom 156 | . . . . . . . . 9 |
16 | 15 | necon3d 2329 | . . . . . . . 8 |
17 | 5, 16 | syl5 32 | . . . . . . 7 |
18 | 17 | imp 123 | . . . . . 6 |
19 | eldifsn 3620 | . . . . . 6 | |
20 | 4, 18, 19 | sylanbrc 413 | . . . . 5 |
21 | 20 | ex 114 | . . . 4 |
22 | 21 | ss2rabdv 3148 | . . 3 |
23 | eqid 2117 | . . . 4 | |
24 | 23 | mptpreima 5002 | . . 3 |
25 | eqid 2117 | . . . 4 | |
26 | 25 | mptpreima 5002 | . . 3 |
27 | 22, 24, 26 | 3sstr4g 3110 | . 2 |
28 | suppssov1.s | . 2 | |
29 | 27, 28 | sstrd 3077 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wne 2285 wral 2393 crab 2397 cvv 2660 cdif 3038 wss 3041 csn 3497 cmpt 3959 ccnv 4508 cima 4512 (class class class)co 5742 |
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-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-xp 4515 df-rel 4516 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: suppssof1 5967 |
Copyright terms: Public domain | W3C validator |