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Mirrors > Home > NFE Home > Th. List > pwpw0 | Unicode version |
Description: Compute the power set of the power set of the empty set. (See pw0 4160 for the power set of the empty set.) Theorem 90 of [Suppes] p. 48. Although this theorem is a special case of pwsn 3881, we have chosen to show a direct elementary proof. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
pwpw0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3262 | . . . . . . . . 9 | |
2 | elsn 3748 | . . . . . . . . . . 11 | |
3 | 2 | imbi2i 303 | . . . . . . . . . 10 |
4 | 3 | albii 1566 | . . . . . . . . 9 |
5 | 1, 4 | bitri 240 | . . . . . . . 8 |
6 | neq0 3560 | . . . . . . . . . 10 | |
7 | exintr 1614 | . . . . . . . . . 10 | |
8 | 6, 7 | syl5bi 208 | . . . . . . . . 9 |
9 | exancom 1586 | . . . . . . . . . . 11 | |
10 | df-clel 2349 | . . . . . . . . . . 11 | |
11 | 9, 10 | bitr4i 243 | . . . . . . . . . 10 |
12 | snssi 3852 | . . . . . . . . . 10 | |
13 | 11, 12 | sylbi 187 | . . . . . . . . 9 |
14 | 8, 13 | syl6 29 | . . . . . . . 8 |
15 | 5, 14 | sylbi 187 | . . . . . . 7 |
16 | 15 | anc2li 540 | . . . . . 6 |
17 | eqss 3287 | . . . . . 6 | |
18 | 16, 17 | syl6ibr 218 | . . . . 5 |
19 | 18 | orrd 367 | . . . 4 |
20 | 0ss 3579 | . . . . . 6 | |
21 | sseq1 3292 | . . . . . 6 | |
22 | 20, 21 | mpbiri 224 | . . . . 5 |
23 | eqimss 3323 | . . . . 5 | |
24 | 22, 23 | jaoi 368 | . . . 4 |
25 | 19, 24 | impbii 180 | . . 3 |
26 | 25 | abbii 2465 | . 2 |
27 | df-pw 3724 | . 2 | |
28 | dfpr2 3749 | . 2 | |
29 | 26, 27, 28 | 3eqtr4i 2383 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 357 wa 358 wal 1540 wex 1541 wceq 1642 wcel 1710 cab 2339 wss 3257 c0 3550 cpw 3722 csn 3737 cpr 3738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-pr 3742 |
This theorem is referenced by: (None) |
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