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Theorem fin23lem38 9373
Description: Lemma for fin23 9413. 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 7233 . . . . . . . 8 (𝑑 ∈ ω → suc 𝑑 ∈ ω)
2 eqid 2771 . . . . . . . . . 10 ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)
3 fveq2 6332 . . . . . . . . . . . . . 14 (𝑏 = suc 𝑑 → (𝑌𝑏) = (𝑌‘suc 𝑑))
43rneqd 5491 . . . . . . . . . . . . 13 (𝑏 = suc 𝑑 → ran (𝑌𝑏) = ran (𝑌‘suc 𝑑))
54unieqd 4584 . . . . . . . . . . . 12 (𝑏 = suc 𝑑 ran (𝑌𝑏) = ran (𝑌‘suc 𝑑))
65eqeq2d 2781 . . . . . . . . . . 11 (𝑏 = suc 𝑑 → ( ran (𝑌‘suc 𝑑) = ran (𝑌𝑏) ↔ ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)))
76rspcev 3460 . . . . . . . . . 10 ((suc 𝑑 ∈ ω ∧ ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)) → ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
82, 7mpan2 671 . . . . . . . . 9 (suc 𝑑 ∈ ω → ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
9 fvex 6342 . . . . . . . . . . . 12 (𝑌‘suc 𝑑) ∈ V
109rnex 7247 . . . . . . . . . . 11 ran (𝑌‘suc 𝑑) ∈ V
1110uniex 7100 . . . . . . . . . 10 ran (𝑌‘suc 𝑑) ∈ V
12 eqid 2771 . . . . . . . . . . 11 (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = (𝑏 ∈ ω ↦ ran (𝑌𝑏))
1312elrnmpt 5510 . . . . . . . . . 10 ( ran (𝑌‘suc 𝑑) ∈ V → ( ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ↔ ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏)))
1411, 13ax-mp 5 . . . . . . . . 9 ( ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ↔ ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
158, 14sylibr 224 . . . . . . . 8 (suc 𝑑 ∈ ω → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
161, 15syl 17 . . . . . . 7 (𝑑 ∈ ω → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
1716adantl 467 . . . . . 6 ((𝜑𝑑 ∈ ω) → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
18 intss1 4626 . . . . . 6 ( ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌‘suc 𝑑))
1917, 18syl 17 . . . . 5 ((𝜑𝑑 ∈ ω) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌‘suc 𝑑))
20 fin23lem33.f . . . . . 6 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
21 fin23lem.f . . . . . 6 (𝜑:ω–1-1→V)
22 fin23lem.g . . . . . 6 (𝜑 ran 𝐺)
23 fin23lem.h . . . . . 6 (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
24 fin23lem.i . . . . . 6 𝑌 = (rec(𝑖, ) ↾ ω)
2520, 21, 22, 23, 24fin23lem35 9371 . . . . 5 ((𝜑𝑑 ∈ ω) → ran (𝑌‘suc 𝑑) ⊊ ran (𝑌𝑑))
2619, 25sspsstrd 3865 . . . 4 ((𝜑𝑑 ∈ ω) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑))
27 dfpss2 3842 . . . . 5 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑) ↔ ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌𝑑) ∧ ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑)))
2827simprbi 484 . . . 4 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑) → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
2926, 28syl 17 . . 3 ((𝜑𝑑 ∈ ω) → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
3029nrexdv 3149 . 2 (𝜑 → ¬ ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
31 fveq2 6332 . . . . . . 7 (𝑏 = 𝑑 → (𝑌𝑏) = (𝑌𝑑))
3231rneqd 5491 . . . . . 6 (𝑏 = 𝑑 → ran (𝑌𝑏) = ran (𝑌𝑑))
3332unieqd 4584 . . . . 5 (𝑏 = 𝑑 ran (𝑌𝑏) = ran (𝑌𝑑))
3433cbvmptv 4884 . . . 4 (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = (𝑑 ∈ ω ↦ ran (𝑌𝑑))
3534elrnmpt 5510 . . 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)))
3635ibi 256 . 2 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) → ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
3730, 36nsyl 137 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 196  wa 382  wal 1629   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  Vcvv 3351  wss 3723  wpss 3724  𝒫 cpw 4297   cuni 4574   cint 4611  cmpt 4863  ran crn 5250  cres 5251  suc csuc 5868  1-1wf1 6028  cfv 6031  (class class class)co 6793  ωcom 7212  reccrdg 7658  𝑚 cmap 8009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659
This theorem is referenced by:  fin23lem39  9374
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