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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 8631 | . . . 4 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 5605 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
3 | pw2eng 8751 | . . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) |
5 | 2onn 8368 | . . . . . 6 ⊢ 2o ∈ ω | |
6 | 5 | elexi 3427 | . . . . 5 ⊢ 2o ∈ V |
7 | 6 | enref 8661 | . . . 4 ⊢ 2o ≈ 2o |
8 | mapen 8810 | . . . 4 ⊢ ((2o ≈ 2o ∧ 𝐴 ≈ 𝐵) → (2o ↑m 𝐴) ≈ (2o ↑m 𝐵)) | |
9 | 7, 8 | mpan 690 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (2o ↑m 𝐴) ≈ (2o ↑m 𝐵)) |
10 | 1 | brrelex2i 5606 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
11 | pw2eng 8751 | . . . 4 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ≈ (2o ↑m 𝐵)) | |
12 | ensym 8677 | . . . 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 8680 | . . 3 ⊢ (((2o ↑m 𝐴) ≈ (2o ↑m 𝐵) ∧ (2o ↑m 𝐵) ≈ 𝒫 𝐵) → (2o ↑m 𝐴) ≈ 𝒫 𝐵) | |
15 | 9, 13, 14 | syl2anc 587 | . 2 ⊢ (𝐴 ≈ 𝐵 → (2o ↑m 𝐴) ≈ 𝒫 𝐵) |
16 | entr 8680 | . 2 ⊢ ((𝒫 𝐴 ≈ (2o ↑m 𝐴) ∧ (2o ↑m 𝐴) ≈ 𝒫 𝐵) → 𝒫 𝐴 ≈ 𝒫 𝐵) | |
17 | 4, 15, 16 | syl2anc 587 | 1 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3408 𝒫 cpw 4513 class class class wbr 5053 (class class class)co 7213 ωcom 7644 2oc2o 8196 ↑m cmap 8508 ≈ cen 8623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-en 8627 |
This theorem is referenced by: pwfiOLD 8971 dfac12k 9761 pwdjuidm 9805 pwsdompw 9818 ackbij2lem2 9854 engch 10242 gchdomtri 10243 canthp1lem1 10266 gchdjuidm 10282 gchxpidm 10283 gchpwdom 10284 gchhar 10293 inar1 10389 rexpen 15789 enrelmap 41282 enrelmapr 41283 |
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