Theorem uniwf 9707

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 9664	. . . . . . . 8 ⊢ Tr (𝑅1‘suc (rank‘𝐴))
2		rankidb 9688	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
3		trss 5203	. . . . . . . 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 9689	. . . . . . . 8 ⊢ (rank‘𝐴) ∈ dom 𝑅1
6		r1sucg 9657	. . . . . . . 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 3970	. . . . . 6 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
9		sspwuni 5043	. . . . . 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 6830	. . . . . 6 ⊢ (𝑅1‘(rank‘𝐴)) ∈ V
12	11	elpw2 5267	. . . . 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 2842	. . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
15		r1elwf 9684	. . 3 ⊢ (∪ 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
16	14, 15	syl 17	. 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
17		pwwf 9695	. . 3 ⊢ (∪ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 ∪ 𝐴 ∈ ∪ (𝑅1 “ On))
18		pwuni 4891	. . . 4 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
19		sswf 9696	. . . 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 1541   ∈ wcel 2111   ⊆ wss 3897  𝒫 cpw 4545  ∪ cuni 4854  Tr wtr 5193  dom cdm 5611   “ cima 5614  Oncon0 6301  suc csuc 6303  ‘cfv 6476  𝑅₁cr1 9650  rankcrnk 9651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-r1 9652  df-rank 9653

This theorem is referenced by:  rankuni2b  9741  r1limwun  10622  wfgru  10702  elwf  35100  dmwf  44998  rnwf  44999  wfaxun  45032  wfac8prim  45035