Theorem uniwf 7701

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 7658	. . . . . . . 8 |- Tr ( R1 ` suc ( rank ` A ) )
2		rankidb 7682	. . . . . . . 8 |- ( A e. U. ( R1 " On ) -> A e. ( R1 ` suc ( rank ` A ) ) )
3		trss 4271	. . . . . . . 8 |- ( Tr ( R1 ` suc ( rank ` A ) ) -> ( A e. ( R1 ` suc ( rank ` A ) ) -> A C_ ( R1 ` suc ( rank ` A ) ) ) )
4	1, 2, 3	mpsyl 61	. . . . . . 7 |- ( A e. U. ( R1 " On ) -> A C_ ( R1 ` suc ( rank ` A ) ) )
5		rankdmr1 7683	. . . . . . . 8 |- ( rank ` A ) e. dom R1
6		r1sucg 7651	. . . . . . . 8 |- ( ( rank ` A ) e. dom R1 -> ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) ) )
7	5, 6	ax-mp 8	. . . . . . 7 |- ( R1 ` suc ( rank ` A ) ) = ~P ( R1 ` ( rank ` A ) )
8	4, 7	syl6sseq 3354	. . . . . 6 |- ( A e. U. ( R1 " On ) -> A C_ ~P ( R1 ` ( rank ` A ) ) )
9		sspwuni 4136	. . . . . 6 |- ( A C_ ~P ( R1 ` ( rank ` A ) ) <-> U. A C_ ( R1 ` ( rank ` A ) ) )
10	8, 9	sylib 189	. . . . 5 |- ( A e. U. ( R1 " On ) -> U. A C_ ( R1 ` ( rank ` A ) ) )
11		fvex 5701	. . . . . 6 |- ( R1 ` ( rank ` A ) ) e. _V
12	11	elpw2 4324	. . . . 5 |- ( U. A e. ~P ( R1 ` ( rank ` A ) ) <-> U. A C_ ( R1 ` ( rank ` A ) ) )
13	10, 12	sylibr 204	. . . 4 |- ( A e. U. ( R1 " On ) -> U. A e. ~P ( R1 ` ( rank ` A ) ) )
14	13, 7	syl6eleqr 2495	. . 3 |- ( A e. U. ( R1 " On ) -> U. A e. ( R1 ` suc ( rank ` A ) ) )
15		r1elwf 7678	. . 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 7689	. . 3 |- ( U. A e. U. ( R1 " On ) <-> ~P U. A e. U. ( R1 " On ) )
18		pwuni 4355	. . . 4 |- A C_ ~P U. A
19		sswf 7690	. . . 4 |- ( ( ~P U. A e. U. ( R1 " On ) /\ A C_ ~P U. A ) -> A e. U. ( R1 " On ) )
20	18, 19	mpan2 653	. . 3 |- ( ~P U. A e. U. ( R1 " On ) -> A e. U. ( R1 " On ) )
21	17, 20	sylbi 188	. 2 |- ( U. A e. U. ( R1 " On ) -> A e. U. ( R1 " On ) )
22	16, 21	impbii 181	1 |- ( A e. U. ( R1 " On ) <-> U. A e. U. ( R1 " On ) )

Syntax hints:   <-> wb 177   = wceq 1649   e. wcel 1721   C_ wss 3280   ~Pcpw 3759   U.cuni 3975   Tr wtr 4262   Oncon0 4541   suc csuc 4543   dom cdm 4837   "cima 4840   ` cfv 5413   R1cr1 7644   rankcrnk 7645

This theorem is referenced by:  rankuni2b  7735  r1limwun  8567  wfgru  8647

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627  df-r1 7646  df-rank 7647