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 2741 | . . . . . . . 8 | |
3 | 1, 2 | syl 14 | . . . . . . 7 |
4 | 3 | adantr 274 | . . . . . 6 |
5 | eldifsni 3710 | . . . . . . . 8 | |
6 | oveq2 5859 | . . . . . . . . . . . 12 | |
7 | 6 | eqeq1d 2179 | . . . . . . . . . . 11 |
8 | suppssov1.o | . . . . . . . . . . . . 13 | |
9 | 8 | ralrimiva 2543 | . . . . . . . . . . . 12 |
10 | 9 | adantr 274 | . . . . . . . . . . 11 |
11 | suppssov1.b | . . . . . . . . . . 11 | |
12 | 7, 10, 11 | rspcdva 2839 | . . . . . . . . . 10 |
13 | oveq1 5858 | . . . . . . . . . . 11 | |
14 | 13 | eqeq1d 2179 | . . . . . . . . . 10 |
15 | 12, 14 | syl5ibrcom 156 | . . . . . . . . 9 |
16 | 15 | necon3d 2384 | . . . . . . . 8 |
17 | 5, 16 | syl5 32 | . . . . . . 7 |
18 | 17 | imp 123 | . . . . . 6 |
19 | eldifsn 3708 | . . . . . 6 | |
20 | 4, 18, 19 | sylanbrc 415 | . . . . 5 |
21 | 20 | ex 114 | . . . 4 |
22 | 21 | ss2rabdv 3228 | . . 3 |
23 | eqid 2170 | . . . 4 | |
24 | 23 | mptpreima 5102 | . . 3 |
25 | eqid 2170 | . . . 4 | |
26 | 25 | mptpreima 5102 | . . 3 |
27 | 22, 24, 26 | 3sstr4g 3190 | . 2 |
28 | suppssov1.s | . 2 | |
29 | 27, 28 | sstrd 3157 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wne 2340 wral 2448 crab 2452 cvv 2730 cdif 3118 wss 3121 csn 3581 cmpt 4048 ccnv 4608 cima 4612 (class class class)co 5851 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-xp 4615 df-rel 4616 df-cnv 4617 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fv 5204 df-ov 5854 |
This theorem is referenced by: suppssof1 6076 |
Copyright terms: Public domain | W3C validator |