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Theorem pw2f1o 7213
 Description: The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.)
Hypotheses
Ref Expression
pw2f1o.1
pw2f1o.2
pw2f1o.3
pw2f1o.4
pw2f1o.5
Assertion
Ref Expression
pw2f1o
Distinct variable groups:   ,,   ,,   ,,   ,
Allowed substitution hints:   ()   (,)   (,)   (,)

Proof of Theorem pw2f1o
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pw2f1o.5 . 2
2 eqid 2436 . . . 4
3 pw2f1o.1 . . . . . 6
4 pw2f1o.2 . . . . . 6
5 pw2f1o.3 . . . . . 6
6 pw2f1o.4 . . . . . 6
73, 4, 5, 6pw2f1olem 7212 . . . . 5
87biimpa 471 . . . 4
92, 8mpanr2 666 . . 3
109simpld 446 . 2
11 vex 2959 . . . 4
1211cnvex 5406 . . 3
13 imaexg 5217 . . 3
1412, 13mp1i 12 . 2
153, 4, 5, 6pw2f1olem 7212 . 2
161, 10, 14, 15f1od 6294 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725   wne 2599  cvv 2956  cif 3739  cpw 3799  csn 3814  cpr 3815   cmpt 4266  ccnv 4877  cima 4881  wf1o 5453  (class class class)co 6081   cmap 7018 This theorem is referenced by:  pw2eng  7214  indf1o  24421 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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020
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