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Mirrors > Home > ILE Home > Th. List > sbthlem2 | Unicode version |
Description: Lemma for isbth 6941. (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 6931 | . . . . . . . 8 |
4 | imass2 4985 | . . . . . . . 8 | |
5 | sscon 3261 | . . . . . . . 8 | |
6 | 3, 4, 5 | mp2b 8 | . . . . . . 7 |
7 | imass2 4985 | . . . . . . 7 | |
8 | sscon 3261 | . . . . . . 7 | |
9 | 6, 7, 8 | mp2b 8 | . . . . . 6 |
10 | imassrn 4962 | . . . . . . . 8 | |
11 | sstr2 3154 | . . . . . . . 8 | |
12 | 10, 11 | ax-mp 5 | . . . . . . 7 |
13 | difss 3253 | . . . . . . 7 | |
14 | ssconb 3260 | . . . . . . 7 | |
15 | 12, 13, 14 | sylancl 411 | . . . . . 6 |
16 | 9, 15 | mpbiri 167 | . . . . 5 |
17 | 16, 13 | jctil 310 | . . . 4 |
18 | 1, 13 | ssexi 4125 | . . . . 5 |
19 | sseq1 3170 | . . . . . 6 | |
20 | imaeq2 4947 | . . . . . . . . 9 | |
21 | 20 | difeq2d 3245 | . . . . . . . 8 |
22 | 21 | imaeq2d 4951 | . . . . . . 7 |
23 | difeq2 3239 | . . . . . . 7 | |
24 | 22, 23 | sseq12d 3178 | . . . . . 6 |
25 | 19, 24 | anbi12d 470 | . . . . 5 |
26 | 18, 25 | elab 2874 | . . . 4 |
27 | 17, 26 | sylibr 133 | . . 3 |
28 | 27, 2 | eleqtrrdi 2264 | . 2 |
29 | elssuni 3822 | . 2 | |
30 | 28, 29 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 cab 2156 cvv 2730 cdif 3118 wss 3121 cuni 3794 crn 4610 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-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 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: sbthlemi3 6933 |
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