Theorem uniwf 8308

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 8265	. . . . . . . 8 |- Tr ( R1 ` suc ( rank ` A ) )
2		rankidb 8289	. . . . . . . 8 |- ( A e. U. ( R1 " On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
3		trss 4499	. . . . . . . 8 |- ( Tr ( R1 ` suc ( rank ` A ) ) -> ( A e. ( R1 ` suc ( rank ` A ) ) -> A C_ ( R1 ` suc ( rank ` A ) ) ) )
4	1, 2, 3	mpsyl 64	. . . . . . 7 |- ( A e. U. ( R1 " On ) -> A C_ ( R1 ` suc ( rank ` A ) ) )
5		rankdmr1 8290	. . . . . . . 8 |- ( rank ` A ) e. dom R1
6		r1sucg 8258	. . . . . . . 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 3464	. . . . . 6 |- ( A e. U. ( R1 " On ) -> A C_ ~P ( R1 ` ( rank ` A ) ) )
9		sspwuni 4360	. . . . . 6 |- ( A C_ ~P ( R1 ` ( rank ` A ) ) <-> U. A C_ ( R1 ` ( rank ` A ) ) )
10	8, 9	sylib 201	. . . . 5 |- ( A e. U. ( R1 " On ) -> U. A C_ ( R1 ` ( rank ` A ) ) )
11		fvex 5889	. . . . . 6 |- ( R1 ` ( rank ` A ) ) e. _V
12	11	elpw2 4565	. . . . 5 |- ( U. A e. ~P ( R1 ` ( rank ` A ) ) <-> U. A C_ ( R1 ` ( rank ` A ) ) )
13	10, 12	sylibr 217	. . . 4 |- ( A e. U. ( R1 " On ) -> U. A e. ~P ( R1 ` ( rank ` A ) ) )
14	13, 7	syl6eleqr 2560	. . 3 |- ( A e. U. ( R1 " On ) -> U. A e. ( R1 ` suc ( rank ` A ) ) )
15		r1elwf 8285	. . 3 |- ( U. A e. ( R1 ` suc ( rank ` A ) ) -> U. A e. U. ( R1 " On ) )
16	14, 15	syl 17	. 2 |- ( A e. U. ( R1 " On ) -> U. A e. U. ( R1 " On ) )
17		pwwf 8296	. . 3 |- ( U. A e. U. ( R1 " On ) <-> ~P U. A e. U. ( R1 " On ) )
18		pwuni 4631	. . . 4 |- A C_ ~P U. A
19		sswf 8297	. . . 4 |- ( ( ~P U. A e. U. ( R1 " On ) /\ A C_ ~P U. A ) -> A e. U. ( R1 " On ) )
20	18, 19	mpan2 685	. . 3 |- ( ~P U. A e. U. ( R1 " On ) -> A e. U. ( R1 " On ) )
21	17, 20	sylbi 200	. 2 |- ( U. A e. U. ( R1 " On ) -> A e. U. ( R1 " On ) )
22	16, 21	impbii 192	1 |- ( A e. U. ( R1 " On ) <-> U. A e. U. ( R1 " On ) )

Syntax hints:   <-> wb 189   = wceq 1452   e. wcel 1904   C_ wss 3390   ~Pcpw 3942   U.cuni 4190   Tr wtr 4490   dom cdm 4839   "cima 4842   Oncon0 5430   suc csuc 5432   ` cfv 5589   R1cr1 8251   rankcrnk 8252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-r1 8253  df-rank 8254

This theorem is referenced by:  rankuni2b  8342  r1limwun  9179  wfgru  9259