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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poss 4294 |
. . 3
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2 | ssel 3149 |
. . . . . . . 8
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3 | ssel 3149 |
. . . . . . . 8
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4 | ssel 3149 |
. . . . . . . 8
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5 | 2, 3, 4 | 3anim123d 1319 |
. . . . . . 7
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6 | 5 | imim1d 75 |
. . . . . 6
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7 | 6 | 2alimdv 1881 |
. . . . 5
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8 | 7 | alimdv 1879 |
. . . 4
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9 | r3al 2521 |
. . . 4
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10 | r3al 2521 |
. . . 4
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11 | 8, 9, 10 | 3imtr4g 205 |
. . 3
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12 | 1, 11 | anim12d 335 |
. 2
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13 | df-iso 4293 |
. 2
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14 | df-iso 4293 |
. 2
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15 | 12, 13, 14 | 3imtr4g 205 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-in 3135 df-ss 3142 df-po 4292 df-iso 4293 |
This theorem is referenced by: soeq2 4312 |
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