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Mirrors > Home > ILE Home > Th. List > sbthlem2 | Unicode version |
Description: Lemma for isbth 6923. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
sbthlem.1 | |
sbthlem.2 |
Ref | Expression |
---|---|
sbthlem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbthlem.1 | . . . . . . . . 9 | |
2 | sbthlem.2 | . . . . . . . . 9 | |
3 | 1, 2 | sbthlem1 6913 | . . . . . . . 8 |
4 | imass2 4974 | . . . . . . . 8 | |
5 | sscon 3251 | . . . . . . . 8 | |
6 | 3, 4, 5 | mp2b 8 | . . . . . . 7 |
7 | imass2 4974 | . . . . . . 7 | |
8 | sscon 3251 | . . . . . . 7 | |
9 | 6, 7, 8 | mp2b 8 | . . . . . 6 |
10 | imassrn 4951 | . . . . . . . 8 | |
11 | sstr2 3144 | . . . . . . . 8 | |
12 | 10, 11 | ax-mp 5 | . . . . . . 7 |
13 | difss 3243 | . . . . . . 7 | |
14 | ssconb 3250 | . . . . . . 7 | |
15 | 12, 13, 14 | sylancl 410 | . . . . . 6 |
16 | 9, 15 | mpbiri 167 | . . . . 5 |
17 | 16, 13 | jctil 310 | . . . 4 |
18 | 1, 13 | ssexi 4114 | . . . . 5 |
19 | sseq1 3160 | . . . . . 6 | |
20 | imaeq2 4936 | . . . . . . . . 9 | |
21 | 20 | difeq2d 3235 | . . . . . . . 8 |
22 | 21 | imaeq2d 4940 | . . . . . . 7 |
23 | difeq2 3229 | . . . . . . 7 | |
24 | 22, 23 | sseq12d 3168 | . . . . . 6 |
25 | 19, 24 | anbi12d 465 | . . . . 5 |
26 | 18, 25 | elab 2865 | . . . 4 |
27 | 17, 26 | sylibr 133 | . . 3 |
28 | 27, 2 | eleqtrrdi 2258 | . 2 |
29 | elssuni 3811 | . 2 | |
30 | 28, 29 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wcel 2135 cab 2150 cvv 2721 cdif 3108 wss 3111 cuni 3783 crn 4599 cima 4601 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 |
This theorem is referenced by: sbthlemi3 6915 |
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