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Theorem uniwf 8959
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 8916 . . . . . . . 8 Tr (𝑅1‘suc (rank‘𝐴))
2 rankidb 8940 . . . . . . . 8 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
3 trss 4984 . . . . . . . 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 8941 . . . . . . . 8 (rank‘𝐴) ∈ dom 𝑅1
6 r1sucg 8909 . . . . . . . 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, 7syl6sseq 3876 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
9 sspwuni 4832 . . . . . 6 (𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
108, 9sylib 210 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
11 fvex 6446 . . . . . 6 (𝑅1‘(rank‘𝐴)) ∈ V
1211elpw2 5050 . . . . 5 ( 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
1310, 12sylibr 226 . . . 4 (𝐴 (𝑅1 “ On) → 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
1413, 7syl6eleqr 2917 . . 3 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
15 r1elwf 8936 . . 3 ( 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
1614, 15syl 17 . 2 (𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
17 pwwf 8947 . . 3 ( 𝐴 (𝑅1 “ On) ↔ 𝒫 𝐴 (𝑅1 “ On))
18 pwuni 4696 . . . 4 𝐴 ⊆ 𝒫 𝐴
19 sswf 8948 . . . 4 ((𝒫 𝐴 (𝑅1 “ On) ∧ 𝐴 ⊆ 𝒫 𝐴) → 𝐴 (𝑅1 “ On))
2018, 19mpan2 684 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2117, 20sylbi 209 . 2 ( 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2216, 21impbii 201 1 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1658  wcel 2166  wss 3798  𝒫 cpw 4378   cuni 4658  Tr wtr 4975  dom cdm 5342  cima 5345  Oncon0 5963  suc csuc 5965  cfv 6123  𝑅1cr1 8902  rankcrnk 8903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-om 7327  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-r1 8904  df-rank 8905
This theorem is referenced by:  rankuni2b  8993  r1limwun  9873  wfgru  9953
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