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Mirrors > Home > ILE Home > Th. List > soss | Unicode version |
Description: Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
soss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poss 4035 | . . 3 | |
2 | ssel 2939 | . . . . . . . 8 | |
3 | ssel 2939 | . . . . . . . 8 | |
4 | ssel 2939 | . . . . . . . 8 | |
5 | 2, 3, 4 | 3anim123d 1214 | . . . . . . 7 |
6 | 5 | imim1d 69 | . . . . . 6 |
7 | 6 | 2alimdv 1761 | . . . . 5 |
8 | 7 | alimdv 1759 | . . . 4 |
9 | r3al 2366 | . . . 4 | |
10 | r3al 2366 | . . . 4 | |
11 | 8, 9, 10 | 3imtr4g 194 | . . 3 |
12 | 1, 11 | anim12d 318 | . 2 |
13 | df-iso 4034 | . 2 | |
14 | df-iso 4034 | . 2 | |
15 | 12, 13, 14 | 3imtr4g 194 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 w3a 885 wal 1241 wcel 1393 wral 2306 wss 2917 class class class wbr 3764 wpo 4031 wor 4032 |
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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
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-in 2924 df-ss 2931 df-po 4033 df-iso 4034 |
This theorem is referenced by: soeq2 4053 |
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