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Theorem pwen 8084
Description: If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
pwen (𝐴𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵)

Proof of Theorem pwen
StepHypRef Expression
1 relen 7911 . . . 4 Rel ≈
21brrelexi 5123 . . 3 (𝐴𝐵𝐴 ∈ V)
3 pw2eng 8017 . . 3 (𝐴 ∈ V → 𝒫 𝐴 ≈ (2𝑜𝑚 𝐴))
42, 3syl 17 . 2 (𝐴𝐵 → 𝒫 𝐴 ≈ (2𝑜𝑚 𝐴))
5 2onn 7672 . . . . . 6 2𝑜 ∈ ω
65elexi 3202 . . . . 5 2𝑜 ∈ V
76enref 7939 . . . 4 2𝑜 ≈ 2𝑜
8 mapen 8075 . . . 4 ((2𝑜 ≈ 2𝑜𝐴𝐵) → (2𝑜𝑚 𝐴) ≈ (2𝑜𝑚 𝐵))
97, 8mpan 705 . . 3 (𝐴𝐵 → (2𝑜𝑚 𝐴) ≈ (2𝑜𝑚 𝐵))
101brrelex2i 5124 . . . 4 (𝐴𝐵𝐵 ∈ V)
11 pw2eng 8017 . . . 4 (𝐵 ∈ V → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
12 ensym 7956 . . . 4 (𝒫 𝐵 ≈ (2𝑜𝑚 𝐵) → (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵)
1310, 11, 123syl 18 . . 3 (𝐴𝐵 → (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵)
14 entr 7959 . . 3 (((2𝑜𝑚 𝐴) ≈ (2𝑜𝑚 𝐵) ∧ (2𝑜𝑚 𝐵) ≈ 𝒫 𝐵) → (2𝑜𝑚 𝐴) ≈ 𝒫 𝐵)
159, 13, 14syl2anc 692 . 2 (𝐴𝐵 → (2𝑜𝑚 𝐴) ≈ 𝒫 𝐵)
16 entr 7959 . 2 ((𝒫 𝐴 ≈ (2𝑜𝑚 𝐴) ∧ (2𝑜𝑚 𝐴) ≈ 𝒫 𝐵) → 𝒫 𝐴 ≈ 𝒫 𝐵)
174, 15, 16syl2anc 692 1 (𝐴𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Vcvv 3189  𝒫 cpw 4135   class class class wbr 4618  (class class class)co 6610  ωcom 7019  2𝑜c2o 7506  𝑚 cmap 7809  cen 7903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-1o 7512  df-2o 7513  df-er 7694  df-map 7811  df-en 7907
This theorem is referenced by:  pwfi  8212  dfac12k  8920  pwcdaidm  8968  pwsdompw  8977  ackbij2lem2  9013  engch  9401  gchdomtri  9402  canthp1lem1  9425  gchcdaidm  9441  gchxpidm  9442  gchpwdom  9443  gchhar  9452  inar1  9548  rexpen  14889  enrelmap  37800  enrelmapr  37801
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