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Mirrors > Home > MPE Home > Th. List > pwen | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
pwen | ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 7846 | . . . 4 ⊢ Rel ≈ | |
2 | 1 | brrelexi 5082 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
3 | pw2eng 7951 | . . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴)) |
5 | 2onn 7607 | . . . . . 6 ⊢ 2𝑜 ∈ ω | |
6 | 5 | elexi 3186 | . . . . 5 ⊢ 2𝑜 ∈ V |
7 | 6 | enref 7874 | . . . 4 ⊢ 2𝑜 ≈ 2𝑜 |
8 | mapen 8009 | . . . 4 ⊢ ((2𝑜 ≈ 2𝑜 ∧ 𝐴 ≈ 𝐵) → (2𝑜 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 𝐵)) | |
9 | 7, 8 | mpan 702 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (2𝑜 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 𝐵)) |
10 | 1 | brrelex2i 5083 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
11 | pw2eng 7951 | . . . 4 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵)) | |
12 | ensym 7891 | . . . 4 ⊢ (𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵) → (2𝑜 ↑𝑚 𝐵) ≈ 𝒫 𝐵) | |
13 | 10, 11, 12 | 3syl 18 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (2𝑜 ↑𝑚 𝐵) ≈ 𝒫 𝐵) |
14 | entr 7894 | . . 3 ⊢ (((2𝑜 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 𝐵) ∧ (2𝑜 ↑𝑚 𝐵) ≈ 𝒫 𝐵) → (2𝑜 ↑𝑚 𝐴) ≈ 𝒫 𝐵) | |
15 | 9, 13, 14 | syl2anc 691 | . 2 ⊢ (𝐴 ≈ 𝐵 → (2𝑜 ↑𝑚 𝐴) ≈ 𝒫 𝐵) |
16 | entr 7894 | . 2 ⊢ ((𝒫 𝐴 ≈ (2𝑜 ↑𝑚 𝐴) ∧ (2𝑜 ↑𝑚 𝐴) ≈ 𝒫 𝐵) → 𝒫 𝐴 ≈ 𝒫 𝐵) | |
17 | 4, 15, 16 | syl2anc 691 | 1 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 𝒫 cpw 4108 class class class wbr 4583 (class class class)co 6549 ωcom 6957 2𝑜c2o 7441 ↑𝑚 cmap 7744 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-1o 7447 df-2o 7448 df-er 7629 df-map 7746 df-en 7842 |
This theorem is referenced by: pwfi 8144 dfac12k 8852 pwcdaidm 8900 pwsdompw 8909 ackbij2lem2 8945 engch 9329 gchdomtri 9330 canthp1lem1 9353 gchcdaidm 9369 gchxpidm 9370 gchpwdom 9371 gchhar 9380 inar1 9476 rexpen 14796 enrelmap 37311 enrelmapr 37312 |
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