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Mirrors > Home > MPE Home > Th. List > pwwf | Structured version Visualization version GIF version |
Description: A power set is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
pwwf | ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1rankidb 8550 | . . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴))) | |
2 | sspwb 4844 | . . . . . . 7 ⊢ (𝐴 ⊆ (𝑅1‘(rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴))) | |
3 | 1, 2 | sylib 207 | . . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴))) |
4 | rankdmr1 8547 | . . . . . . 7 ⊢ (rank‘𝐴) ∈ dom 𝑅1 | |
5 | r1sucg 8515 | . . . . . . 7 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)) |
7 | 3, 6 | syl6sseqr 3615 | . . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴))) |
8 | fvex 6113 | . . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) ∈ V | |
9 | 8 | elpw2 4755 | . . . . 5 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴))) |
10 | 7, 9 | sylibr 223 | . . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴))) |
11 | r1funlim 8512 | . . . . . . . 8 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
12 | 11 | simpri 477 | . . . . . . 7 ⊢ Lim dom 𝑅1 |
13 | limsuc 6941 | . . . . . . 7 ⊢ (Lim dom 𝑅1 → ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1)) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1) |
15 | 4, 14 | mpbi 219 | . . . . 5 ⊢ suc (rank‘𝐴) ∈ dom 𝑅1 |
16 | r1sucg 8515 | . . . . 5 ⊢ (suc (rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴))) | |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴)) |
18 | 10, 17 | syl6eleqr 2699 | . . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴))) |
19 | r1elwf 8542 | . . 3 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On)) | |
20 | 18, 19 | syl 17 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On)) |
21 | r1elssi 8551 | . . 3 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ ∪ (𝑅1 “ On)) | |
22 | elex 3185 | . . . . 5 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ V) | |
23 | pwexb 6867 | . . . . 5 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | |
24 | 22, 23 | sylibr 223 | . . . 4 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ V) |
25 | pwidg 4121 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴) | |
26 | 24, 25 | syl 17 | . . 3 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴) |
27 | 21, 26 | sseldd 3569 | . 2 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
28 | 20, 27 | impbii 198 | 1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 dom cdm 5038 “ cima 5041 Oncon0 5640 Lim wlim 5641 suc csuc 5642 Fun wfun 5798 ‘cfv 5804 𝑅1cr1 8508 rankcrnk 8509 |
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-reu 2903 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 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-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 df-rank 8511 |
This theorem is referenced by: snwf 8555 uniwf 8565 rankpwi 8569 r1pw 8591 r1pwcl 8593 dfac12r 8851 wfgru 9517 |
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