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Mirrors > Home > MPE Home > Th. List > pwpw0 | Structured version Visualization version Unicode version |
Description: Compute the power set of the power set of the empty set. (See pw0 4343 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 4428, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
pwpw0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3591 | . . . . . . . . 9 | |
2 | velsn 4193 | . . . . . . . . . . 11 | |
3 | 2 | imbi2i 326 | . . . . . . . . . 10 |
4 | 3 | albii 1747 | . . . . . . . . 9 |
5 | 1, 4 | bitri 264 | . . . . . . . 8 |
6 | neq0 3930 | . . . . . . . . . 10 | |
7 | exintr 1819 | . . . . . . . . . 10 | |
8 | 6, 7 | syl5bi 232 | . . . . . . . . 9 |
9 | exancom 1787 | . . . . . . . . . . 11 | |
10 | df-clel 2618 | . . . . . . . . . . 11 | |
11 | 9, 10 | bitr4i 267 | . . . . . . . . . 10 |
12 | snssi 4339 | . . . . . . . . . 10 | |
13 | 11, 12 | sylbi 207 | . . . . . . . . 9 |
14 | 8, 13 | syl6 35 | . . . . . . . 8 |
15 | 5, 14 | sylbi 207 | . . . . . . 7 |
16 | 15 | anc2li 580 | . . . . . 6 |
17 | eqss 3618 | . . . . . 6 | |
18 | 16, 17 | syl6ibr 242 | . . . . 5 |
19 | 18 | orrd 393 | . . . 4 |
20 | 0ss 3972 | . . . . . 6 | |
21 | sseq1 3626 | . . . . . 6 | |
22 | 20, 21 | mpbiri 248 | . . . . 5 |
23 | eqimss 3657 | . . . . 5 | |
24 | 22, 23 | jaoi 394 | . . . 4 |
25 | 19, 24 | impbii 199 | . . 3 |
26 | 25 | abbii 2739 | . 2 |
27 | df-pw 4160 | . 2 | |
28 | dfpr2 4195 | . 2 | |
29 | 26, 27, 28 | 3eqtr4i 2654 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 383 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 cab 2608 wss 3574 c0 3915 cpw 4158 csn 4177 cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 |
This theorem is referenced by: pp0ex 4855 pwcda1 9016 canthp1lem1 9474 rankeq1o 32278 ssoninhaus 32447 |
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