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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 8989 | . . . 4 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 5745 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
3 | pw2eng 9117 | . . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) |
5 | 2onn 8679 | . . . . . 6 ⊢ 2o ∈ ω | |
6 | 5 | elexi 3501 | . . . . 5 ⊢ 2o ∈ V |
7 | 6 | enref 9024 | . . . 4 ⊢ 2o ≈ 2o |
8 | mapen 9180 | . . . 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 5746 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
11 | pw2eng 9117 | . . . 4 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ≈ (2o ↑m 𝐵)) | |
12 | ensym 9042 | . . . 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 9045 | . . 3 ⊢ (((2o ↑m 𝐴) ≈ (2o ↑m 𝐵) ∧ (2o ↑m 𝐵) ≈ 𝒫 𝐵) → (2o ↑m 𝐴) ≈ 𝒫 𝐵) | |
15 | 9, 13, 14 | syl2anc 584 | . 2 ⊢ (𝐴 ≈ 𝐵 → (2o ↑m 𝐴) ≈ 𝒫 𝐵) |
16 | entr 9045 | . 2 ⊢ ((𝒫 𝐴 ≈ (2o ↑m 𝐴) ∧ (2o ↑m 𝐴) ≈ 𝒫 𝐵) → 𝒫 𝐴 ≈ 𝒫 𝐵) | |
17 | 4, 15, 16 | syl2anc 584 | 1 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3478 𝒫 cpw 4605 class class class wbr 5148 (class class class)co 7431 ωcom 7887 2oc2o 8499 ↑m cmap 8865 ≈ cen 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 |
This theorem is referenced by: dfac12k 10186 pwdjuidm 10230 pwsdompw 10241 ackbij2lem2 10277 engch 10666 gchdomtri 10667 canthp1lem1 10690 gchdjuidm 10706 gchxpidm 10707 gchpwdom 10708 gchhar 10717 inar1 10813 rexpen 16261 enrelmap 43987 enrelmapr 43988 |
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