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Mirrors > Home > ILE Home > Th. List > tposss | Unicode version |
Description: Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposss | tpos tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coss1 4491 | . . 3 | |
2 | dmss 4534 | . . . . . 6 | |
3 | cnvss 4508 | . . . . . 6 | |
4 | unss1 3112 | . . . . . 6 | |
5 | resmpt 4656 | . . . . . 6 | |
6 | 2, 3, 4, 5 | 4syl 18 | . . . . 5 |
7 | resss 4635 | . . . . 5 | |
8 | 6, 7 | syl6eqssr 2996 | . . . 4 |
9 | coss2 4492 | . . . 4 | |
10 | 8, 9 | syl 14 | . . 3 |
11 | 1, 10 | sstrd 2955 | . 2 |
12 | df-tpos 5860 | . 2 tpos | |
13 | df-tpos 5860 | . 2 tpos | |
14 | 11, 12, 13 | 3sstr4g 2986 | 1 tpos tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1243 cun 2915 wss 2917 c0 3224 csn 3375 cuni 3580 cmpt 3818 ccnv 4344 cdm 4345 cres 4347 ccom 4349 tpos ctpos 5859 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-mpt 3820 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-tpos 5860 |
This theorem is referenced by: tposeq 5862 |
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