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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 4190 | . . 3 | |
2 | ssel 3061 | . . . . . . . 8 | |
3 | ssel 3061 | . . . . . . . 8 | |
4 | ssel 3061 | . . . . . . . 8 | |
5 | 2, 3, 4 | 3anim123d 1282 | . . . . . . 7 |
6 | 5 | imim1d 75 | . . . . . 6 |
7 | 6 | 2alimdv 1837 | . . . . 5 |
8 | 7 | alimdv 1835 | . . . 4 |
9 | r3al 2454 | . . . 4 | |
10 | r3al 2454 | . . . 4 | |
11 | 8, 9, 10 | 3imtr4g 204 | . . 3 |
12 | 1, 11 | anim12d 333 | . 2 |
13 | df-iso 4189 | . 2 | |
14 | df-iso 4189 | . 2 | |
15 | 12, 13, 14 | 3imtr4g 204 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 w3a 947 wal 1314 wcel 1465 wral 2393 wss 3041 class class class wbr 3899 wpo 4186 wor 4187 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-in 3047 df-ss 3054 df-po 4188 df-iso 4189 |
This theorem is referenced by: soeq2 4208 |
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