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Mathbox for Alan Sare |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sspwimp | Structured version Visualization version Unicode version |
Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. sspwimp 37315, using conventional notation, was translated from virtual deduction form, sspwimpVD 37316, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspwimp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3048 |
. . . . . . 7
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2 | 1 | a1i 11 |
. . . . . 6
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3 | id 22 |
. . . . . . 7
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4 | id 22 |
. . . . . . . 8
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5 | elpwi 3960 |
. . . . . . . 8
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6 | 4, 5 | syl 17 |
. . . . . . 7
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7 | sstr 3440 |
. . . . . . . 8
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8 | 7 | ancoms 455 |
. . . . . . 7
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9 | 3, 6, 8 | syl2an 480 |
. . . . . 6
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10 | 2, 9 | elpwgded 36931 |
. . . . . 6
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11 | 2, 9, 10 | uun0.1 37165 |
. . . . 5
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12 | 11 | ex 436 |
. . . 4
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13 | 12 | alrimiv 1773 |
. . 3
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14 | dfss2 3421 |
. . . 4
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15 | 14 | biimpri 210 |
. . 3
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16 | 13, 15 | syl 17 |
. 2
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17 | 16 | iin1 36942 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 |
This theorem depends on definitions: df-bi 189 df-an 373 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-v 3047 df-in 3411 df-ss 3418 df-pw 3953 |
This theorem is referenced by: (None) |
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