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Mirrors > Home > ILE Home > Th. List > sspwb | Unicode version |
Description: Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.) |
Ref | Expression |
---|---|
sspwb |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 2946 |
. . . . 5
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2 | 1 | com12 27 |
. . . 4
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3 | vex 2554 |
. . . . 5
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4 | 3 | elpw 3357 |
. . . 4
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5 | 3 | elpw 3357 |
. . . 4
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6 | 2, 4, 5 | 3imtr4g 194 |
. . 3
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7 | 6 | ssrdv 2945 |
. 2
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8 | ssel 2933 |
. . . 4
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9 | snexgOLD 3926 |
. . . . . . 7
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10 | 3, 9 | ax-mp 7 |
. . . . . 6
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11 | 10 | elpw 3357 |
. . . . 5
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12 | 3 | snss 3485 |
. . . . 5
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13 | 11, 12 | bitr4i 176 |
. . . 4
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14 | 10 | elpw 3357 |
. . . . 5
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15 | 3 | snss 3485 |
. . . . 5
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16 | 14, 15 | bitr4i 176 |
. . . 4
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17 | 8, 13, 16 | 3imtr3g 193 |
. . 3
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18 | 17 | ssrdv 2945 |
. 2
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19 | 7, 18 | impbii 117 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 |
This theorem is referenced by: pwel 3945 ssextss 3947 pweqb 3950 |
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