Theorem uniwf 9860

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 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On))

Proof of Theorem uniwf

Step	Hyp	Ref	Expression
1		r1tr 9817	. . . . . . . 8 ⊢ Tr (𝑅1‘suc (rank‘𝐴))
2		rankidb 9841	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
3		trss 5269	. . . . . . . 8 ⊢ (Tr (𝑅1‘suc (rank‘𝐴)) → (𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴))))
4	1, 2, 3	mpsyl 68	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
5		rankdmr1 9842	. . . . . . . 8 ⊢ (rank‘𝐴) ∈ dom 𝑅1
6		r1sucg 9810	. . . . . . . 8 ⊢ ((rank‘𝐴) ∈ 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	sseqtrdi 4023	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
9		sspwuni 5099	. . . . . 6 ⊢ (𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ ∪ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
10	8, 9	sylib 218	. . . . 5 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
11		fvex 6918	. . . . . 6 ⊢ (𝑅1‘(rank‘𝐴)) ∈ V
12	11	elpw2 5333	. . . . 5 ⊢ (∪ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ ∪ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
13	10, 12	sylibr 234	. . . 4 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
14	13, 7	eleqtrrdi 2851	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
15		r1elwf 9837	. . 3 ⊢ (∪ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
16	14, 15	syl 17	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
17		pwwf 9848	. . 3 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
18		pwuni 4944	. . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
19		sswf 9849	. . . 4 ⊢ ((𝒫 ∪ 𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On))
20	18, 19	mpan2 691	. . 3 ⊢ (𝒫 ∪ 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
21	17, 20	sylbi 217	. 2 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ ∪ (𝑅1 “ On))
22	16, 21	impbii 209	1 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝐴 ∈ ∪ (𝑅1 “ On))

Syntax hints:   ↔ wb 206   = wceq 1539   ∈ wcel 2107   ⊆ wss 3950  𝒫 cpw 4599  ∪ cuni 4906  Tr wtr 5258  dom cdm 5684   “ cima 5687  Oncon0 6383  suc csuc 6385  ‘cfv 6560  𝑅₁cr1 9803  rankcrnk 9804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-r1 9805  df-rank 9806

This theorem is referenced by:  rankuni2b  9894  r1limwun  10777  wfgru  10857  dmwf  44987  rnwf  44988  wfaxun  45021  wfac8prim  45024