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Mirrors > Home > MPE Home > Th. List > pwdju1 | Structured version Visualization version GIF version |
Description: The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
pwdju1 | ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ 𝒫 (𝐴 ⊔ 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 8420 | . . . . 5 ⊢ 1o ∈ On | |
2 | pwdjuen 10113 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1o ∈ On) → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 × 𝒫 1o)) | |
3 | 1, 2 | mpan2 689 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 × 𝒫 1o)) |
4 | pwexg 5331 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
5 | 1oex 8418 | . . . . . 6 ⊢ 1o ∈ V | |
6 | 5 | pwex 5333 | . . . . 5 ⊢ 𝒫 1o ∈ V |
7 | xpcomeng 9004 | . . . . 5 ⊢ ((𝒫 𝐴 ∈ V ∧ 𝒫 1o ∈ V) → (𝒫 𝐴 × 𝒫 1o) ≈ (𝒫 1o × 𝒫 𝐴)) | |
8 | 4, 6, 7 | sylancl 586 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 × 𝒫 1o) ≈ (𝒫 1o × 𝒫 𝐴)) |
9 | entr 8942 | . . . 4 ⊢ ((𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 × 𝒫 1o) ∧ (𝒫 𝐴 × 𝒫 1o) ≈ (𝒫 1o × 𝒫 𝐴)) → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 1o × 𝒫 𝐴)) | |
10 | 3, 8, 9 | syl2anc 584 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 1o × 𝒫 𝐴)) |
11 | pwpw0 4771 | . . . . . 6 ⊢ 𝒫 {∅} = {∅, {∅}} | |
12 | df1o2 8415 | . . . . . . 7 ⊢ 1o = {∅} | |
13 | 12 | pweqi 4574 | . . . . . 6 ⊢ 𝒫 1o = 𝒫 {∅} |
14 | df2o2 8417 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
15 | 11, 13, 14 | 3eqtr4i 2774 | . . . . 5 ⊢ 𝒫 1o = 2o |
16 | 15 | xpeq1i 5657 | . . . 4 ⊢ (𝒫 1o × 𝒫 𝐴) = (2o × 𝒫 𝐴) |
17 | xp2dju 10108 | . . . 4 ⊢ (2o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴) | |
18 | 16, 17 | eqtri 2764 | . . 3 ⊢ (𝒫 1o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴) |
19 | 10, 18 | breqtrdi 5144 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴)) |
20 | 19 | ensymd 8941 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ 𝒫 (𝐴 ⊔ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3443 ∅c0 4280 𝒫 cpw 4558 {csn 4584 {cpr 4586 class class class wbr 5103 × cxp 5629 Oncon0 6315 1oc1o 8401 2oc2o 8402 ≈ cen 8876 ⊔ cdju 9830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-1o 8408 df-2o 8409 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-dju 9833 |
This theorem is referenced by: pwdjuidm 10123 djulepw 10124 pwsdompw 10136 gchdjuidm 10600 gchpwdom 10602 |
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