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Mirrors > Home > ILE Home > Th. List > sbthlem1 | Unicode version |
Description: Lemma for isbth 6855. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
sbthlem.1 | |
sbthlem.2 |
Ref | Expression |
---|---|
sbthlem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unissb 3766 | . 2 | |
2 | sbthlem.2 | . . . . 5 | |
3 | 2 | abeq2i 2250 | . . . 4 |
4 | difss2 3204 | . . . . . . 7 | |
5 | ssconb 3209 | . . . . . . . 8 | |
6 | 5 | exbiri 379 | . . . . . . 7 |
7 | 4, 6 | syl5 32 | . . . . . 6 |
8 | 7 | pm2.43d 50 | . . . . 5 |
9 | 8 | imp 123 | . . . 4 |
10 | 3, 9 | sylbi 120 | . . 3 |
11 | elssuni 3764 | . . . . 5 | |
12 | imass2 4915 | . . . . 5 | |
13 | sscon 3210 | . . . . 5 | |
14 | 11, 12, 13 | 3syl 17 | . . . 4 |
15 | imass2 4915 | . . . 4 | |
16 | sscon 3210 | . . . 4 | |
17 | 14, 15, 16 | 3syl 17 | . . 3 |
18 | 10, 17 | sstrd 3107 | . 2 |
19 | 1, 18 | mprgbir 2490 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cab 2125 cvv 2686 cdif 3068 wss 3071 cuni 3736 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 |
This theorem is referenced by: sbthlem2 6846 sbthlemi3 6847 sbthlemi5 6849 |
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