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Theorem fin23lem38 9762
 Description: Lemma for fin23 9802. 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 7584 . . . . . . . 8 (𝑑 ∈ ω → suc 𝑑 ∈ ω)
2 eqid 2798 . . . . . . . . . 10 ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)
3 fveq2 6645 . . . . . . . . . . . . 13 (𝑏 = suc 𝑑 → (𝑌𝑏) = (𝑌‘suc 𝑑))
43rneqd 5772 . . . . . . . . . . . 12 (𝑏 = suc 𝑑 → ran (𝑌𝑏) = ran (𝑌‘suc 𝑑))
54unieqd 4814 . . . . . . . . . . 11 (𝑏 = suc 𝑑 ran (𝑌𝑏) = ran (𝑌‘suc 𝑑))
65rspceeqv 3586 . . . . . . . . . 10 ((suc 𝑑 ∈ ω ∧ ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)) → ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
72, 6mpan2 690 . . . . . . . . 9 (suc 𝑑 ∈ ω → ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
8 fvex 6658 . . . . . . . . . . . 12 (𝑌‘suc 𝑑) ∈ V
98rnex 7601 . . . . . . . . . . 11 ran (𝑌‘suc 𝑑) ∈ V
109uniex 7449 . . . . . . . . . 10 ran (𝑌‘suc 𝑑) ∈ V
11 eqid 2798 . . . . . . . . . . 11 (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = (𝑏 ∈ ω ↦ ran (𝑌𝑏))
1211elrnmpt 5792 . . . . . . . . . 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 237 . . . . . . . 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 485 . . . . . 6 ((𝜑𝑑 ∈ ω) → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
17 intss1 4853 . . . . . 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 9760 . . . . 5 ((𝜑𝑑 ∈ ω) → ran (𝑌‘suc 𝑑) ⊊ ran (𝑌𝑑))
2518, 24sspsstrd 4036 . . . 4 ((𝜑𝑑 ∈ ω) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑))
26 dfpss2 4013 . . . . 5 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑) ↔ ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌𝑑) ∧ ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑)))
2726simprbi 500 . . . 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 3229 . 2 (𝜑 → ¬ ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
30 fveq2 6645 . . . . . . 7 (𝑏 = 𝑑 → (𝑌𝑏) = (𝑌𝑑))
3130rneqd 5772 . . . . . 6 (𝑏 = 𝑑 → ran (𝑌𝑏) = ran (𝑌𝑑))
3231unieqd 4814 . . . . 5 (𝑏 = 𝑑 ran (𝑌𝑏) = ran (𝑌𝑑))
3332cbvmptv 5133 . . . 4 (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = (𝑑 ∈ ω ↦ ran (𝑌𝑑))
3433elrnmpt 5792 . . 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 270 . 2 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) → ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
3629, 35nsyl 142 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 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881   ⊊ wpss 3882  𝒫 cpw 4497  ∪ cuni 4800  ∩ cint 4838   ↦ cmpt 5110  ran crn 5520   ↾ cres 5521  suc csuc 6161  –1-1→wf1 6321  ‘cfv 6324  (class class class)co 7135  ωcom 7562  reccrdg 8030   ↑m cmap 8391 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7563  df-wrecs 7932  df-recs 7993  df-rdg 8031 This theorem is referenced by:  fin23lem39  9763
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