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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 7454 | . . . 4 ⊢ 1𝑜 ∈ On | |
2 | pwcdaen 8890 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 1𝑜 ∈ On) → 𝒫 (𝐴 +𝑐 1𝑜) ≈ (𝒫 𝐴 × 𝒫 1𝑜)) | |
3 | 1, 2 | mpan2 703 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 +𝑐 1𝑜) ≈ (𝒫 𝐴 × 𝒫 1𝑜)) |
4 | pwpw0 4284 | . . . . . 6 ⊢ 𝒫 {∅} = {∅, {∅}} | |
5 | df1o2 7459 | . . . . . . 7 ⊢ 1𝑜 = {∅} | |
6 | 5 | pweqi 4112 | . . . . . 6 ⊢ 𝒫 1𝑜 = 𝒫 {∅} |
7 | df2o2 7461 | . . . . . 6 ⊢ 2𝑜 = {∅, {∅}} | |
8 | 4, 6, 7 | 3eqtr4i 2642 | . . . . 5 ⊢ 𝒫 1𝑜 = 2𝑜 |
9 | 8 | xpeq2i 5060 | . . . 4 ⊢ (𝒫 𝐴 × 𝒫 1𝑜) = (𝒫 𝐴 × 2𝑜) |
10 | pwexg 4776 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
11 | xp2cda 8885 | . . . . 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 2656 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 × 𝒫 1𝑜) = (𝒫 𝐴 +𝑐 𝒫 𝐴)) |
14 | 3, 13 | breqtrd 4609 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 +𝑐 1𝑜) ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴)) |
15 | 14 | ensymd 7893 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 (𝐴 +𝑐 1𝑜)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {cpr 4127 class class class wbr 4583 × cxp 5036 Oncon0 5640 (class class class)co 6549 1𝑜c1o 7440 2𝑜c2o 7441 ≈ cen 7838 +𝑐 ccda 8872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-1o 7447 df-2o 7448 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-cda 8873 |
This theorem is referenced by: pwcdaidm 8900 cdalepw 8901 pwsdompw 8909 gchcdaidm 9369 gchpwdom 9371 |
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