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Mirrors > Home > ILE Home > Th. List > sbthlemi5 | Unicode version |
Description: Lemma for isbth 6823. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
sbthlem.1 | |
sbthlem.2 | |
sbthlem.3 |
Ref | Expression |
---|---|
sbthlemi5 | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbthlem.1 | . . . . . . . . . 10 | |
2 | sbthlem.2 | . . . . . . . . . 10 | |
3 | 1, 2 | sbthlem1 6813 | . . . . . . . . 9 |
4 | difss 3172 | . . . . . . . . 9 | |
5 | 3, 4 | sstri 3076 | . . . . . . . 8 |
6 | sseq2 3091 | . . . . . . . 8 | |
7 | 5, 6 | mpbiri 167 | . . . . . . 7 |
8 | dfss 3055 | . . . . . . 7 | |
9 | 7, 8 | sylib 121 | . . . . . 6 |
10 | 9 | uneq1d 3199 | . . . . 5 |
11 | 1, 2 | sbthlemi3 6815 | . . . . . . . 8 EXMID |
12 | imassrn 4862 | . . . . . . . 8 | |
13 | 11, 12 | eqsstrrdi 3120 | . . . . . . 7 EXMID |
14 | dfss 3055 | . . . . . . 7 | |
15 | 13, 14 | sylib 121 | . . . . . 6 EXMID |
16 | 15 | uneq2d 3200 | . . . . 5 EXMID |
17 | 10, 16 | sylan9eq 2170 | . . . 4 EXMID |
18 | 17 | an12s 539 | . . 3 EXMID |
19 | sbthlem.3 | . . . . 5 | |
20 | 19 | dmeqi 4710 | . . . 4 |
21 | dmun 4716 | . . . 4 | |
22 | dmres 4810 | . . . . 5 | |
23 | dmres 4810 | . . . . . 6 | |
24 | df-rn 4520 | . . . . . . . 8 | |
25 | 24 | eqcomi 2121 | . . . . . . 7 |
26 | 25 | ineq2i 3244 | . . . . . 6 |
27 | 23, 26 | eqtri 2138 | . . . . 5 |
28 | 22, 27 | uneq12i 3198 | . . . 4 |
29 | 20, 21, 28 | 3eqtri 2142 | . . 3 |
30 | 18, 29 | syl6reqr 2169 | . 2 EXMID |
31 | exmidexmid 4090 | . . . . . . 7 EXMID DECID | |
32 | 31 | ralrimivw 2483 | . . . . . 6 EXMID DECID |
33 | 32 | biantrud 302 | . . . . 5 EXMID DECID |
34 | undifdcss 6779 | . . . . 5 DECID | |
35 | 33, 34 | syl6rbbr 198 | . . . 4 EXMID |
36 | 5, 35 | mpbiri 167 | . . 3 EXMID |
37 | 36 | adantr 274 | . 2 EXMID |
38 | 30, 37 | eqtr4d 2153 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 804 wceq 1316 wcel 1465 cab 2103 wral 2393 cvv 2660 cdif 3038 cun 3039 cin 3040 wss 3041 cuni 3706 EXMIDwem 4088 ccnv 4508 cdm 4509 crn 4510 cres 4511 cima 4512 |
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-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-stab 801 df-dc 805 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-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-exmid 4089 df-xp 4515 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 |
This theorem is referenced by: sbthlemi9 6821 |
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