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Mirrors > Home > ILE Home > Th. List > sbthlemi5 | Unicode version |
Description: Lemma for isbth 6855. (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 6845 | . . . . . . . . 9 |
4 | difss 3202 | . . . . . . . . 9 | |
5 | 3, 4 | sstri 3106 | . . . . . . . 8 |
6 | sseq2 3121 | . . . . . . . 8 | |
7 | 5, 6 | mpbiri 167 | . . . . . . 7 |
8 | dfss 3085 | . . . . . . 7 | |
9 | 7, 8 | sylib 121 | . . . . . 6 |
10 | 9 | uneq1d 3229 | . . . . 5 |
11 | 1, 2 | sbthlemi3 6847 | . . . . . . . 8 EXMID |
12 | imassrn 4892 | . . . . . . . 8 | |
13 | 11, 12 | eqsstrrdi 3150 | . . . . . . 7 EXMID |
14 | dfss 3085 | . . . . . . 7 | |
15 | 13, 14 | sylib 121 | . . . . . 6 EXMID |
16 | 15 | uneq2d 3230 | . . . . 5 EXMID |
17 | 10, 16 | sylan9eq 2192 | . . . 4 EXMID |
18 | 17 | an12s 554 | . . 3 EXMID |
19 | sbthlem.3 | . . . . 5 | |
20 | 19 | dmeqi 4740 | . . . 4 |
21 | dmun 4746 | . . . 4 | |
22 | dmres 4840 | . . . . 5 | |
23 | dmres 4840 | . . . . . 6 | |
24 | df-rn 4550 | . . . . . . . 8 | |
25 | 24 | eqcomi 2143 | . . . . . . 7 |
26 | 25 | ineq2i 3274 | . . . . . 6 |
27 | 23, 26 | eqtri 2160 | . . . . 5 |
28 | 22, 27 | uneq12i 3228 | . . . 4 |
29 | 20, 21, 28 | 3eqtri 2164 | . . 3 |
30 | 18, 29 | syl6reqr 2191 | . 2 EXMID |
31 | exmidexmid 4120 | . . . . . . 7 EXMID DECID | |
32 | 31 | ralrimivw 2506 | . . . . . 6 EXMID DECID |
33 | 32 | biantrud 302 | . . . . 5 EXMID DECID |
34 | undifdcss 6811 | . . . . 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 2175 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 819 wceq 1331 wcel 1480 cab 2125 wral 2416 cvv 2686 cdif 3068 cun 3069 cin 3070 wss 3071 cuni 3736 EXMIDwem 4118 ccnv 4538 cdm 4539 crn 4540 cres 4541 cima 4542 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-exmid 4119 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 |
This theorem is referenced by: sbthlemi9 6853 |
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