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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 8499 | . . . . 5 ⊢ 1o ∈ On | |
2 | pwdjuen 10205 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1o ∈ On) → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 × 𝒫 1o)) | |
3 | 1, 2 | mpan2 690 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 × 𝒫 1o)) |
4 | pwexg 5378 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
5 | 1oex 8497 | . . . . . 6 ⊢ 1o ∈ V | |
6 | 5 | pwex 5380 | . . . . 5 ⊢ 𝒫 1o ∈ V |
7 | xpcomeng 9089 | . . . . 5 ⊢ ((𝒫 𝐴 ∈ V ∧ 𝒫 1o ∈ V) → (𝒫 𝐴 × 𝒫 1o) ≈ (𝒫 1o × 𝒫 𝐴)) | |
8 | 4, 6, 7 | sylancl 585 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 × 𝒫 1o) ≈ (𝒫 1o × 𝒫 𝐴)) |
9 | entr 9027 | . . . 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 583 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 1o × 𝒫 𝐴)) |
11 | pwpw0 4817 | . . . . . 6 ⊢ 𝒫 {∅} = {∅, {∅}} | |
12 | df1o2 8494 | . . . . . . 7 ⊢ 1o = {∅} | |
13 | 12 | pweqi 4619 | . . . . . 6 ⊢ 𝒫 1o = 𝒫 {∅} |
14 | df2o2 8496 | . . . . . 6 ⊢ 2o = {∅, {∅}} | |
15 | 11, 13, 14 | 3eqtr4i 2766 | . . . . 5 ⊢ 𝒫 1o = 2o |
16 | 15 | xpeq1i 5704 | . . . 4 ⊢ (𝒫 1o × 𝒫 𝐴) = (2o × 𝒫 𝐴) |
17 | xp2dju 10200 | . . . 4 ⊢ (2o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴) | |
18 | 16, 17 | eqtri 2756 | . . 3 ⊢ (𝒫 1o × 𝒫 𝐴) = (𝒫 𝐴 ⊔ 𝒫 𝐴) |
19 | 10, 18 | breqtrdi 5189 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 ⊔ 1o) ≈ (𝒫 𝐴 ⊔ 𝒫 𝐴)) |
20 | 19 | ensymd 9026 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ 𝒫 (𝐴 ⊔ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 Vcvv 3471 ∅c0 4323 𝒫 cpw 4603 {csn 4629 {cpr 4631 class class class wbr 5148 × cxp 5676 Oncon0 6369 1oc1o 8480 2oc2o 8481 ≈ cen 8961 ⊔ cdju 9922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-dju 9925 |
This theorem is referenced by: pwdjuidm 10215 djulepw 10216 pwsdompw 10228 gchdjuidm 10692 gchpwdom 10694 |
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