MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniwf Structured version   Visualization version   GIF version

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
StepHypRef 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))))
41, 2, 3mpsyl 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)))
75, 6ax-mp 5 . . . . . . 7 (𝑅1‘suc (rank‘𝐴)) = 𝒫 (𝑅1‘(rank‘𝐴))
84, 7sseqtrdi 4023 . . . . . 6 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)))
9 sspwuni 5099 . . . . . 6 (𝐴 ⊆ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
108, 9sylib 218 . . . . 5 (𝐴 (𝑅1 “ On) → 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
11 fvex 6918 . . . . . 6 (𝑅1‘(rank‘𝐴)) ∈ V
1211elpw2 5333 . . . . 5 ( 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)) ↔ 𝐴 ⊆ (𝑅1‘(rank‘𝐴)))
1310, 12sylibr 234 . . . 4 (𝐴 (𝑅1 “ On) → 𝐴 ∈ 𝒫 (𝑅1‘(rank‘𝐴)))
1413, 7eleqtrrdi 2851 . . 3 (𝐴 (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)))
15 r1elwf 9837 . . 3 ( 𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐴 (𝑅1 “ On))
1614, 15syl 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))
2018, 19mpan2 691 . . 3 (𝒫 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2117, 20sylbi 217 . 2 ( 𝐴 (𝑅1 “ On) → 𝐴 (𝑅1 “ On))
2216, 21impbii 209 1 (𝐴 (𝑅1 “ On) ↔ 𝐴 (𝑅1 “ On))
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator