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Theorem uniwf 9844
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
StepHypRef Expression
1 r1tr 9801 . . . . . . . 8 Tr (𝑅1‘suc (rank‘𝐴))
2 rankidb 9825 . . . . . . . 8 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
3 trss 5277 . . . . . . . 8 (Tr (𝑅1‘suc (rank‘𝐴)) → (𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴))))
41, 2, 3mpsyl 68 . . . . . . 7 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘suc (rank‘𝐴)))
5 rankdmr1 9826 . . . . . . . 8 (rank‘𝐴) ∈ dom 𝑅1
6 r1sucg 9794 . . . . . . . 8 ((rank‘𝐴) ∈ dom 𝑅1 → (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴)))
75, 6ax-mp 5 . . . . . . 7 (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
84, 7sseqtrdi 4027 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
9 sspwuni 5104 . . . . . 6 (𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
108, 9sylib 217 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
11 fvex 6909 . . . . . 6 (𝑅1‘(rank‘𝐴)) ∈ V
1211elpw2 5348 . . . . 5 ( 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
1310, 12sylibr 233 . . . 4 (𝐴 (𝑅1 “ On) → 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
1413, 7eleqtrrdi 2836 . . 3 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
15 r1elwf 9821 . . 3 ( 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
1614, 15syl 17 . 2 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
17 pwwf 9832 . . 3 ( 𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
18 pwuni 4949 . . . 4 𝐴 ⊆ 𝒫 𝐴
19 sswf 9833 . . . 4 ((𝒫 𝐴 (𝑅1 “ On) ∧ 𝐴 ⊆ 𝒫 𝐴) → 𝐴 (𝑅1 “ On))
2018, 19mpan2 689 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2117, 20sylbi 216 . 2 ( 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2216, 21impbii 208 1 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  wss 3944  𝒫 cpw 4604   cuni 4909  Tr wtr 5266  dom cdm 5678  cima 5681  Oncon0 6371  suc csuc 6373  cfv 6549  𝑅1cr1 9787  rankcrnk 9788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-r1 9789  df-rank 9790
This theorem is referenced by:  rankuni2b  9878  r1limwun  10761  wfgru  10841
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