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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 8891 | . . . 4 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 5689 | . . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
3 | pw2eng 9025 | . . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ (2o ↑m 𝐴)) |
5 | 2onn 8589 | . . . . . 6 ⊢ 2o ∈ ω | |
6 | 5 | elexi 3463 | . . . . 5 ⊢ 2o ∈ V |
7 | 6 | enref 8928 | . . . 4 ⊢ 2o ≈ 2o |
8 | mapen 9088 | . . . 4 ⊢ ((2o ≈ 2o ∧ 𝐴 ≈ 𝐵) → (2o ↑m 𝐴) ≈ (2o ↑m 𝐵)) | |
9 | 7, 8 | mpan 689 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (2o ↑m 𝐴) ≈ (2o ↑m 𝐵)) |
10 | 1 | brrelex2i 5690 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
11 | pw2eng 9025 | . . . 4 ⊢ (𝐵 ∈ V → 𝒫 𝐵 ≈ (2o ↑m 𝐵)) | |
12 | ensym 8946 | . . . 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 8949 | . . 3 ⊢ (((2o ↑m 𝐴) ≈ (2o ↑m 𝐵) ∧ (2o ↑m 𝐵) ≈ 𝒫 𝐵) → (2o ↑m 𝐴) ≈ 𝒫 𝐵) | |
15 | 9, 13, 14 | syl2anc 585 | . 2 ⊢ (𝐴 ≈ 𝐵 → (2o ↑m 𝐴) ≈ 𝒫 𝐵) |
16 | entr 8949 | . 2 ⊢ ((𝒫 𝐴 ≈ (2o ↑m 𝐴) ∧ (2o ↑m 𝐴) ≈ 𝒫 𝐵) → 𝒫 𝐴 ≈ 𝒫 𝐵) | |
17 | 4, 15, 16 | syl2anc 585 | 1 ⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3444 𝒫 cpw 4561 class class class wbr 5106 (class class class)co 7358 ωcom 7803 2oc2o 8407 ↑m cmap 8768 ≈ cen 8883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-1o 8413 df-2o 8414 df-er 8651 df-map 8770 df-en 8887 |
This theorem is referenced by: pwfiOLD 9294 dfac12k 10088 pwdjuidm 10132 pwsdompw 10145 ackbij2lem2 10181 engch 10569 gchdomtri 10570 canthp1lem1 10593 gchdjuidm 10609 gchxpidm 10610 gchpwdom 10611 gchhar 10620 inar1 10716 rexpen 16115 enrelmap 42357 enrelmapr 42358 |
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