Theorem pwwf 9815

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.)

Assertion

Ref	Expression
pwwf	⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))

Proof of Theorem pwwf

Step	Hyp	Ref	Expression
1		r1rankidb 9812	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
2	1	sspwd 4569	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
3		rankdmr1 9809	. . . . . . 7 ⊢ (rank‘𝐴) ∈ dom 𝑅1
4		r1sucg 9776	. . . . . . 7 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
6	2, 5	sseqtrrdi 3949	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
7		fvex 6898	. . . . . 6 ⊢ (𝑅1‘suc (rank‘𝐴)) ∈ V
8	7	elpw2 5295	. . . . 5 ⊢ (𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)) ↔ 𝒫 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
9	6, 8	sylibr 237	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ 𝒫 (𝑅1‘suc (rank‘𝐴)))
10		r1dmlim 9772	. . . . . . 7 ⊢ Lim dom 𝑅1
11		limsuc 7860	. . . . . . 7 ⊢ (Lim dom 𝑅1 → ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ ((rank‘𝐴) ∈ dom 𝑅1 ↔ suc (rank‘𝐴) ∈ dom 𝑅1)
13	3, 12	mpbi 233	. . . . 5 ⊢ suc (rank‘𝐴) ∈ dom 𝑅1
14		r1sucg 9776	. . . . 5 ⊢ (suc (rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴)))
15	13, 14	ax-mp 5	. . . 4 ⊢ (𝑅1‘suc suc (rank‘𝐴)) = 𝒫 (𝑅1‘suc (rank‘𝐴))
16	9, 15	eleqtrrdi 2871	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)))
17		r1elwf 9804	. . 3 ⊢ (𝒫 𝐴 ∈ (𝑅1‘suc suc (rank‘𝐴)) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
18	16, 17	syl 18	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))
19		r1elssi 9813	. . 3 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝒫 𝐴 ⊆ ∪ (𝑅1 “ On))
20		pwexr 7779	. . . 4 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ V)
21		pwidg 4576	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ∈ 𝒫 𝐴)
22	20, 21	syl 18	. . 3 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ 𝒫 𝐴)
23	19, 22	sseldd 3909	. 2 ⊢ (𝒫 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
24	18, 23	impbii 212	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝐴 ∈ ∪ (𝑅1 “ On))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895  𝒫 cpw 4556  ∪ cuni 4866  dom cdm 5650   “ cima 5653  Oncon0 6362  Lim wlim 6363  suc csuc 6364  ‘cfv 6538  𝑅₁cr1 9766  rankcrnk 9767

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-r1 9768  df-rank 9769

This theorem is used by:  snwf  9817  uniwf  9828  rankpwi  9832  r1pw  9859  r1pwcl  9861  dfac12r  10225  wfgru  10901  xpwf  45997  wfaxsep  46028  wfaxpow  46030