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Mirrors > Home > ILE Home > Th. List > sbthlem1 | Unicode version |
Description: Lemma for isbth 6942. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
sbthlem.1 | |
sbthlem.2 |
Ref | Expression |
---|---|
sbthlem1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unissb 3824 | . 2 | |
2 | sbthlem.2 | . . . . 5 | |
3 | 2 | abeq2i 2281 | . . . 4 |
4 | difss2 3255 | . . . . . . 7 | |
5 | ssconb 3260 | . . . . . . . 8 | |
6 | 5 | exbiri 380 | . . . . . . 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 3822 | . . . . 5 | |
12 | imass2 4985 | . . . . 5 | |
13 | sscon 3261 | . . . . 5 | |
14 | 11, 12, 13 | 3syl 17 | . . . 4 |
15 | imass2 4985 | . . . 4 | |
16 | sscon 3261 | . . . 4 | |
17 | 14, 15, 16 | 3syl 17 | . . 3 |
18 | 10, 17 | sstrd 3157 | . 2 |
19 | 1, 18 | mprgbir 2528 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cab 2156 cvv 2730 cdif 3118 wss 3121 cuni 3794 cima 4612 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-cnv 4617 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 |
This theorem is referenced by: sbthlem2 6933 sbthlemi3 6934 sbthlemi5 6936 |
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