Theorem uniwf 8014

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 7971	. . . . . . . 8 |- Tr ( R1 ` suc ( rank ` A ) )
2		rankidb 7995	. . . . . . . 8 |- ( A e. U. ( R1 " On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
3		trss 4382	. . . . . . . 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 7996	. . . . . . . 8 |- ( rank ` A ) e. dom R1
6		r1sucg 7964	. . . . . . . 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 3390	. . . . . 6 |- ( A e. U. ( R1 " On ) -> A C_ ~P ( R1 ` ( rank ` A ) ) )
9		sspwuni 4244	. . . . . 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 5689	. . . . . 6 |- ( R1 ` ( rank ` A ) ) e. _V
12	11	elpw2 4444	. . . . 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 2524	. . 3 |- ( A e. U. ( R1 " On ) -> U. A e. ( R1 ` suc ( rank ` A ) ) )
15		r1elwf 7991	. . 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 8002	. . 3 |- ( U. A e. U. ( R1 " On ) <-> ~P U. A e. U. ( R1 " On ) )
18		pwuni 4511	. . . 4 |- A C_ ~P U. A
19		sswf 8003	. . . 4 |- ( ( ~P U. A e. U. ( R1 " On ) /\ A C_ ~P U. A ) -> A e. U. ( R1 " On ) )
20	18, 19	mpan2 664	. . 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 1362   e. wcel 1755   C_ wss 3316   ~Pcpw 3848   U.cuni 4079   Tr wtr 4373   Oncon0 4706   suc csuc 4708   dom cdm 4827   "cima 4830   ` cfv 5406   R1cr1 7957   rankcrnk 7958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-om 6466  df-recs 6818  df-rdg 6852  df-r1 7959  df-rank 7960

This theorem is referenced by:  rankuni2b  8048  r1limwun  8891  wfgru  8971