Theorem uniwf 8228

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 8185	. . . . . . . 8 |- Tr ( R1 ` suc ( rank ` A ) )
2		rankidb 8209	. . . . . . . 8 |- ( A e. U. ( R1 " On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
3		trss 4541	. . . . . . . 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 8210	. . . . . . . 8 |- ( rank ` A ) e. dom R1
6		r1sucg 8178	. . . . . . . 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 3535	. . . . . 6 |- ( A e. U. ( R1 " On ) -> A C_ ~P ( R1 ` ( rank ` A ) ) )
9		sspwuni 4404	. . . . . 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 5858	. . . . . 6 |- ( R1 ` ( rank ` A ) ) e. _V
12	11	elpw2 4601	. . . . 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 2553	. . 3 |- ( A e. U. ( R1 " On ) -> U. A e. ( R1 ` suc ( rank ` A ) ) )
15		r1elwf 8205	. . 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 8216	. . 3 |- ( U. A e. U. ( R1 " On ) <-> ~P U. A e. U. ( R1 " On ) )
18		pwuni 4668	. . . 4 |- A C_ ~P U. A
19		sswf 8217	. . . 4 |- ( ( ~P U. A e. U. ( R1 " On ) /\ A C_ ~P U. A ) -> A e. U. ( R1 " On ) )
20	18, 19	mpan2 669	. . 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 1398   e. wcel 1823   C_ wss 3461   ~Pcpw 3999   U.cuni 4235   Tr wtr 4532   Oncon0 4867   suc csuc 4869   dom cdm 4988   "cima 4991   ` cfv 5570   R1cr1 8171   rankcrnk 8172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173  df-rank 8174

This theorem is referenced by:  rankuni2b  8262  r1limwun  9103  wfgru  9183