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Mirrors > Home > ILE Home > Th. List > sbthlemi6 | Unicode version |
Description: Lemma for isbth 6956. (Contributed by NM, 27-Mar-1998.) |
Ref | Expression |
---|---|
sbthlem.1 | |
sbthlem.2 | |
sbthlem.3 |
Ref | Expression |
---|---|
sbthlemi6 | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 527 | . . 3 EXMID EXMID | |
2 | simprll 537 | . . 3 EXMID | |
3 | simprlr 538 | . . 3 EXMID | |
4 | simprr 531 | . . 3 EXMID | |
5 | rnun 5029 | . . . . 5 | |
6 | sbthlem.3 | . . . . . 6 | |
7 | 6 | rneqi 4848 | . . . . 5 |
8 | df-ima 4633 | . . . . . 6 | |
9 | 8 | uneq1i 3283 | . . . . 5 |
10 | 5, 7, 9 | 3eqtr4i 2206 | . . . 4 |
11 | sbthlem.1 | . . . . . . 7 | |
12 | sbthlem.2 | . . . . . . 7 | |
13 | 11, 12 | sbthlemi4 6949 | . . . . . 6 EXMID |
14 | df-ima 4633 | . . . . . 6 | |
15 | 13, 14 | eqtr3di 2223 | . . . . 5 EXMID |
16 | 15 | uneq2d 3287 | . . . 4 EXMID |
17 | 10, 16 | eqtr4id 2227 | . . 3 EXMID |
18 | 1, 2, 3, 4, 17 | syl121anc 1243 | . 2 EXMID |
19 | imassrn 4974 | . . . . . . 7 | |
20 | sstr2 3160 | . . . . . . 7 | |
21 | 19, 20 | ax-mp 5 | . . . . . 6 |
22 | 21 | adantl 277 | . . . . 5 EXMID |
23 | undifdcss 6912 | . . . . . . 7 DECID | |
24 | exmidexmid 4191 | . . . . . . . . 9 EXMID DECID | |
25 | 24 | ralrimivw 2549 | . . . . . . . 8 EXMID DECID |
26 | 25 | biantrud 304 | . . . . . . 7 EXMID DECID |
27 | 23, 26 | bitr4id 199 | . . . . . 6 EXMID |
28 | 27 | adantr 276 | . . . . 5 EXMID |
29 | 22, 28 | mpbird 167 | . . . 4 EXMID |
30 | 29 | eqcomd 2181 | . . 3 EXMID |
31 | 30 | adantr 276 | . 2 EXMID |
32 | 18, 31 | eqtrd 2208 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 DECID wdc 834 w3a 978 wceq 1353 wcel 2146 cab 2161 wral 2453 cvv 2735 cdif 3124 cun 3125 wss 3127 cuni 3805 EXMIDwem 4189 ccnv 4619 cdm 4620 crn 4621 cres 4622 cima 4623 wfun 5202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-exmid 4190 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-fun 5210 |
This theorem is referenced by: sbthlemi9 6954 |
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