MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem38 Structured version   Visualization version   GIF version

Theorem fin23lem38 9459
Description: Lemma for fin23 9499. The contradictory chain has no minimum. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypotheses
Ref Expression
fin23lem33.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.f (𝜑:ω–1-1→V)
fin23lem.g (𝜑 ran 𝐺)
fin23lem.h (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
fin23lem.i 𝑌 = (rec(𝑖, ) ↾ ω)
Assertion
Ref Expression
fin23lem38 (𝜑 → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
Distinct variable groups:   𝑎,𝑏,𝑔,𝑖,𝑗,𝑥,,𝐺   𝐹,𝑎   𝜑,𝑎,𝑏,𝑗   𝑌,𝑎,𝑏,𝑗
Allowed substitution hints:   𝜑(𝑥,𝑔,,𝑖)   𝐹(𝑥,𝑔,,𝑖,𝑗,𝑏)   𝑌(𝑥,𝑔,,𝑖)

Proof of Theorem fin23lem38
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 peano2 7320 . . . . . . . 8 (𝑑 ∈ ω → suc 𝑑 ∈ ω)
2 eqid 2799 . . . . . . . . . 10 ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)
3 fveq2 6411 . . . . . . . . . . . . 13 (𝑏 = suc 𝑑 → (𝑌𝑏) = (𝑌‘suc 𝑑))
43rneqd 5556 . . . . . . . . . . . 12 (𝑏 = suc 𝑑 → ran (𝑌𝑏) = ran (𝑌‘suc 𝑑))
54unieqd 4638 . . . . . . . . . . 11 (𝑏 = suc 𝑑 ran (𝑌𝑏) = ran (𝑌‘suc 𝑑))
65rspceeqv 3515 . . . . . . . . . 10 ((suc 𝑑 ∈ ω ∧ ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)) → ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
72, 6mpan2 683 . . . . . . . . 9 (suc 𝑑 ∈ ω → ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
8 fvex 6424 . . . . . . . . . . . 12 (𝑌‘suc 𝑑) ∈ V
98rnex 7335 . . . . . . . . . . 11 ran (𝑌‘suc 𝑑) ∈ V
109uniex 7187 . . . . . . . . . 10 ran (𝑌‘suc 𝑑) ∈ V
11 eqid 2799 . . . . . . . . . . 11 (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = (𝑏 ∈ ω ↦ ran (𝑌𝑏))
1211elrnmpt 5576 . . . . . . . . . 10 ( ran (𝑌‘suc 𝑑) ∈ V → ( ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ↔ ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏)))
1310, 12ax-mp 5 . . . . . . . . 9 ( ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ↔ ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
147, 13sylibr 226 . . . . . . . 8 (suc 𝑑 ∈ ω → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
151, 14syl 17 . . . . . . 7 (𝑑 ∈ ω → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
1615adantl 474 . . . . . 6 ((𝜑𝑑 ∈ ω) → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
17 intss1 4682 . . . . . 6 ( ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌‘suc 𝑑))
1816, 17syl 17 . . . . 5 ((𝜑𝑑 ∈ ω) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌‘suc 𝑑))
19 fin23lem33.f . . . . . 6 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
20 fin23lem.f . . . . . 6 (𝜑:ω–1-1→V)
21 fin23lem.g . . . . . 6 (𝜑 ran 𝐺)
22 fin23lem.h . . . . . 6 (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
23 fin23lem.i . . . . . 6 𝑌 = (rec(𝑖, ) ↾ ω)
2419, 20, 21, 22, 23fin23lem35 9457 . . . . 5 ((𝜑𝑑 ∈ ω) → ran (𝑌‘suc 𝑑) ⊊ ran (𝑌𝑑))
2518, 24sspsstrd 3912 . . . 4 ((𝜑𝑑 ∈ ω) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑))
26 dfpss2 3889 . . . . 5 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑) ↔ ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌𝑑) ∧ ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑)))
2726simprbi 491 . . . 4 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑) → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
2825, 27syl 17 . . 3 ((𝜑𝑑 ∈ ω) → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
2928nrexdv 3181 . 2 (𝜑 → ¬ ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
30 fveq2 6411 . . . . . . 7 (𝑏 = 𝑑 → (𝑌𝑏) = (𝑌𝑑))
3130rneqd 5556 . . . . . 6 (𝑏 = 𝑑 → ran (𝑌𝑏) = ran (𝑌𝑑))
3231unieqd 4638 . . . . 5 (𝑏 = 𝑑 ran (𝑌𝑏) = ran (𝑌𝑑))
3332cbvmptv 4943 . . . 4 (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = (𝑑 ∈ ω ↦ ran (𝑌𝑑))
3433elrnmpt 5576 . . 3 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) → ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ↔ ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑)))
3534ibi 259 . 2 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) → ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
3629, 35nsyl 138 1 (𝜑 → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  wal 1651   = wceq 1653  wcel 2157  {cab 2785  wral 3089  wrex 3090  Vcvv 3385  wss 3769  wpss 3770  𝒫 cpw 4349   cuni 4628   cint 4667  cmpt 4922  ran crn 5313  cres 5314  suc csuc 5943  1-1wf1 6098  cfv 6101  (class class class)co 6878  ωcom 7299  reccrdg 7744  𝑚 cmap 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745
This theorem is referenced by:  fin23lem39  9460
  Copyright terms: Public domain W3C validator