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Mirrors > Home > MPE Home > Th. List > pwcda1 | 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 |
---|---|
pwcda1 | ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 (𝐴 +𝑐 1𝑜)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 7567 | . . . 4 ⊢ 1𝑜 ∈ On | |
2 | pwcdaen 9007 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 1𝑜 ∈ On) → 𝒫 (𝐴 +𝑐 1𝑜) ≈ (𝒫 𝐴 × 𝒫 1𝑜)) | |
3 | 1, 2 | mpan2 707 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 +𝑐 1𝑜) ≈ (𝒫 𝐴 × 𝒫 1𝑜)) |
4 | pwpw0 4344 | . . . . . 6 ⊢ 𝒫 {∅} = {∅, {∅}} | |
5 | df1o2 7572 | . . . . . . 7 ⊢ 1𝑜 = {∅} | |
6 | 5 | pweqi 4162 | . . . . . 6 ⊢ 𝒫 1𝑜 = 𝒫 {∅} |
7 | df2o2 7574 | . . . . . 6 ⊢ 2𝑜 = {∅, {∅}} | |
8 | 4, 6, 7 | 3eqtr4i 2654 | . . . . 5 ⊢ 𝒫 1𝑜 = 2𝑜 |
9 | 8 | xpeq2i 5136 | . . . 4 ⊢ (𝒫 𝐴 × 𝒫 1𝑜) = (𝒫 𝐴 × 2𝑜) |
10 | pwexg 4850 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
11 | xp2cda 9002 | . . . . 5 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 × 2𝑜) = (𝒫 𝐴 +𝑐 𝒫 𝐴)) | |
12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 × 2𝑜) = (𝒫 𝐴 +𝑐 𝒫 𝐴)) |
13 | 9, 12 | syl5eq 2668 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 × 𝒫 1𝑜) = (𝒫 𝐴 +𝑐 𝒫 𝐴)) |
14 | 3, 13 | breqtrd 4679 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 +𝑐 1𝑜) ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴)) |
15 | 14 | ensymd 8007 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 (𝐴 +𝑐 1𝑜)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 𝒫 cpw 4158 {csn 4177 {cpr 4179 class class class wbr 4653 × cxp 5112 Oncon0 5723 (class class class)co 6650 1𝑜c1o 7553 2𝑜c2o 7554 ≈ cen 7952 +𝑐 ccda 8989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-1o 7560 df-2o 7561 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-cda 8990 |
This theorem is referenced by: pwcdaidm 9017 cdalepw 9018 pwsdompw 9026 gchcdaidm 9490 gchpwdom 9492 |
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