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Theorem pwsnALT 4011
 Description: The power set of a singleton (direct proof). TO DO - should we keep this? (Contributed by NM, 5-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pwsnALT

Proof of Theorem pwsnALT
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfss2 3338 . . . . . . . . 9
2 elsn 3830 . . . . . . . . . . 11
32imbi2i 305 . . . . . . . . . 10
43albii 1576 . . . . . . . . 9
51, 4bitri 242 . . . . . . . 8
6 neq0 3639 . . . . . . . . . 10
7 exintr 1625 . . . . . . . . . 10
86, 7syl5bi 210 . . . . . . . . 9
9 df-clel 2433 . . . . . . . . . . 11
10 exancom 1597 . . . . . . . . . . 11
119, 10bitr2i 243 . . . . . . . . . 10
12 snssi 3943 . . . . . . . . . 10
1311, 12sylbi 189 . . . . . . . . 9
148, 13syl6 32 . . . . . . . 8
155, 14sylbi 189 . . . . . . 7
1615anc2li 542 . . . . . 6
17 eqss 3364 . . . . . 6
1816, 17syl6ibr 220 . . . . 5
1918orrd 369 . . . 4
20 0ss 3657 . . . . . 6
21 sseq1 3370 . . . . . 6
2220, 21mpbiri 226 . . . . 5
23 eqimss 3401 . . . . 5
2422, 23jaoi 370 . . . 4
2519, 24impbii 182 . . 3
2625abbii 2549 . 2
27 df-pw 3802 . 2
28 dfpr2 3831 . 2
2926, 27, 283eqtr4i 2467 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 359   wa 360  wal 1550  wex 1551   wceq 1653   wcel 1726  cab 2423   wss 3321  c0 3629  cpw 3800  csn 3815  cpr 3816 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-pw 3802  df-sn 3821  df-pr 3822
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