Theorem uniwf 8018

Description: A union is well-founded iff the base set is. (Contributed by Mario Carneiro, 8-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
uniwf	|- ( A e. U. ( R1 " On ) <-> U. A e. U. ( R1 " On ) )

Proof of Theorem uniwf

Step	Hyp	Ref	Expression
1		r1tr 7975	. . . . . . . 8 |- Tr ( R1 ` suc ( rank ` A ) )
2		rankidb 7999	. . . . . . . 8 |- ( A e. U. ( R1 " On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
3		trss 4389	. . . . . . . 8 |- ( Tr ( R1 ` suc ( rank ` A ) ) -> ( A e. ( R1 ` suc ( rank ` A ) ) -> A C_ ( R1 ` suc ( rank ` A ) ) ) )
4	1, 2, 3	mpsyl 63	. . . . . . 7 |- ( A e. U. ( R1 " On ) -> A C_ ( R1 ` suc ( rank ` A ) ) )
5		rankdmr1 8000	. . . . . . . 8 |- ( rank ` A ) e. dom R1
6		r1sucg 7968	. . . . . . . 8 |- ( ( rank ` A ) e. dom R1 -> ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) ) )
7	5, 6	ax-mp 5	. . . . . . 7 |- ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) )
8	4, 7	syl6sseq 3397	. . . . . 6 |- ( A e. U. ( R1 " On ) -> A C_ ~P ( R1 ` ( rank ` A ) ) )
9		sspwuni 4251	. . . . . 6 |- ( A C_ ~P ( R1 ` ( rank ` A ) ) <-> U. A C_ ( R1 ` ( rank ` A ) ) )
10	8, 9	sylib 196	. . . . 5 |- ( A e. U. ( R1 " On ) -> U. A C_ ( R1 ` ( rank ` A ) ) )
11		fvex 5696	. . . . . 6 |- ( R1 ` ( rank ` A ) ) e. _V
12	11	elpw2 4451	. . . . 5 |- ( U. A e. ~P ( R1 ` ( rank ` A ) ) <-> U. A C_ ( R1 ` ( rank ` A ) ) )
13	10, 12	sylibr 212	. . . 4 |- ( A e. U. ( R1 " On ) -> U. A e. ~P ( R1 ` ( rank ` A ) ) )
14	13, 7	syl6eleqr 2529	. . 3 |- ( A e. U. ( R1 " On ) -> U. A e. ( R1 ` suc ( rank ` A ) ) )
15		r1elwf 7995	. . 3 |- ( U. A e. ( R1 ` suc ( rank ` A ) ) -> U. A e. U. ( R1 " On ) )
16	14, 15	syl 16	. 2 |- ( A e. U. ( R1 " On ) -> U. A e. U. ( R1 " On ) )
17		pwwf 8006	. . 3 |- ( U. A e. U. ( R1 " On ) <-> ~P U. A e. U. ( R1 " On ) )
18		pwuni 4518	. . . 4 |- A C_ ~P U. A
19		sswf 8007	. . . 4 |- ( ( ~P U. A e. U. ( R1 " On ) /\ A C_ ~P U. A ) -> A e. U. ( R1 " On ) )
20	18, 19	mpan2 671	. . 3 |- ( ~P U. A e. U. ( R1 " On ) -> A e. U. ( R1 " On ) )
21	17, 20	sylbi 195	. 2 |- ( U. A e. U. ( R1 " On ) -> A e. U. ( R1 " On ) )
22	16, 21	impbii 188	1 |- ( A e. U. ( R1 " On ) <-> U. A e. U. ( R1 " On ) )

Syntax hints:   <-> wb 184   = wceq 1369   e. wcel 1756   C_ wss 3323   ~Pcpw 3855   U.cuni 4086   Tr wtr 4380   Oncon0 4714   suc csuc 4716   dom cdm 4835   "cima 4838   ` cfv 5413   R1cr1 7961   rankcrnk 7962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-om 6472  df-recs 6824  df-rdg 6858  df-r1 7963  df-rank 7964

This theorem is referenced by:  rankuni2b  8052  r1limwun  8895  wfgru  8975