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Theorem pw2eng 7206
 Description: The power set of a set is equinumerous to set exponentiation with a base of ordinal . (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 1-Jul-2015.)
Assertion
Ref Expression
pw2eng

Proof of Theorem pw2eng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4375 . . 3
2 ovex 6098 . . . 4
32a1i 11 . . 3
4 id 20 . . . 4
5 0ex 4331 . . . . 5
65a1i 11 . . . 4
7 p0ex 4378 . . . . 5
87a1i 11 . . . 4
9 0nep0 4362 . . . . 5
109a1i 11 . . . 4
11 eqid 2435 . . . 4
124, 6, 8, 10, 11pw2f1o 7205 . . 3
13 f1oen2g 7116 . . 3
141, 3, 12, 13syl3anc 1184 . 2
15 df2o2 6730 . . 3
1615oveq1i 6083 . 2
1714, 16syl6breqr 4244 1
 Colors of variables: wff set class Syntax hints:   wi 4   wcel 1725   wne 2598  cvv 2948  c0 3620  cif 3731  cpw 3791  csn 3806  cpr 3807   class class class wbr 4204   cmpt 4258  wf1o 5445  (class class class)co 6073  c2o 6710   cmap 7010   cen 7098 This theorem is referenced by:  pw2en  7207  pwen  7272  mappwen  7985  pwcdaen  8057  hauspwdom  17556 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1o 6716  df-2o 6717  df-map 7012  df-en 7102
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