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Mirrors > Home > ILE Home > Th. List > sbthlemi6 | Unicode version |
Description: Lemma for isbth 6923. (Contributed by NM, 27-Mar-1998.) |
Ref | Expression |
---|---|
sbthlem.1 | |
sbthlem.2 | |
sbthlem.3 |
Ref | Expression |
---|---|
sbthlemi6 | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 519 | . . 3 EXMID EXMID | |
2 | simprll 527 | . . 3 EXMID | |
3 | simprlr 528 | . . 3 EXMID | |
4 | simprr 522 | . . 3 EXMID | |
5 | rnun 5006 | . . . . 5 | |
6 | sbthlem.3 | . . . . . 6 | |
7 | 6 | rneqi 4826 | . . . . 5 |
8 | df-ima 4611 | . . . . . 6 | |
9 | 8 | uneq1i 3267 | . . . . 5 |
10 | 5, 7, 9 | 3eqtr4i 2195 | . . . 4 |
11 | sbthlem.1 | . . . . . . 7 | |
12 | sbthlem.2 | . . . . . . 7 | |
13 | 11, 12 | sbthlemi4 6916 | . . . . . 6 EXMID |
14 | df-ima 4611 | . . . . . 6 | |
15 | 13, 14 | eqtr3di 2212 | . . . . 5 EXMID |
16 | 15 | uneq2d 3271 | . . . 4 EXMID |
17 | 10, 16 | eqtr4id 2216 | . . 3 EXMID |
18 | 1, 2, 3, 4, 17 | syl121anc 1232 | . 2 EXMID |
19 | imassrn 4951 | . . . . . . 7 | |
20 | sstr2 3144 | . . . . . . 7 | |
21 | 19, 20 | ax-mp 5 | . . . . . 6 |
22 | 21 | adantl 275 | . . . . 5 EXMID |
23 | undifdcss 6879 | . . . . . . 7 DECID | |
24 | exmidexmid 4169 | . . . . . . . . 9 EXMID DECID | |
25 | 24 | ralrimivw 2538 | . . . . . . . 8 EXMID DECID |
26 | 25 | biantrud 302 | . . . . . . 7 EXMID DECID |
27 | 23, 26 | bitr4id 198 | . . . . . 6 EXMID |
28 | 27 | adantr 274 | . . . . 5 EXMID |
29 | 22, 28 | mpbird 166 | . . . 4 EXMID |
30 | 29 | eqcomd 2170 | . . 3 EXMID |
31 | 30 | adantr 274 | . 2 EXMID |
32 | 18, 31 | eqtrd 2197 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 DECID wdc 824 w3a 967 wceq 1342 wcel 2135 cab 2150 wral 2442 cvv 2721 cdif 3108 cun 3109 wss 3111 cuni 3783 EXMIDwem 4167 ccnv 4597 cdm 4598 crn 4599 cres 4600 cima 4601 wfun 5176 |
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-nul 4102 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 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-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-exmid 4168 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-fun 5184 |
This theorem is referenced by: sbthlemi9 6921 |
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