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Theorem fin23lem38 9617
Description: Lemma for fin23 9657. 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 7458 . . . . . . . 8 (𝑑 ∈ ω → suc 𝑑 ∈ ω)
2 eqid 2795 . . . . . . . . . 10 ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)
3 fveq2 6538 . . . . . . . . . . . . 13 (𝑏 = suc 𝑑 → (𝑌𝑏) = (𝑌‘suc 𝑑))
43rneqd 5690 . . . . . . . . . . . 12 (𝑏 = suc 𝑑 → ran (𝑌𝑏) = ran (𝑌‘suc 𝑑))
54unieqd 4755 . . . . . . . . . . 11 (𝑏 = suc 𝑑 ran (𝑌𝑏) = ran (𝑌‘suc 𝑑))
65rspceeqv 3577 . . . . . . . . . 10 ((suc 𝑑 ∈ ω ∧ ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)) → ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
72, 6mpan2 687 . . . . . . . . 9 (suc 𝑑 ∈ ω → ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
8 fvex 6551 . . . . . . . . . . . 12 (𝑌‘suc 𝑑) ∈ V
98rnex 7473 . . . . . . . . . . 11 ran (𝑌‘suc 𝑑) ∈ V
109uniex 7323 . . . . . . . . . 10 ran (𝑌‘suc 𝑑) ∈ V
11 eqid 2795 . . . . . . . . . . 11 (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = (𝑏 ∈ ω ↦ ran (𝑌𝑏))
1211elrnmpt 5710 . . . . . . . . . 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 235 . . . . . . . 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 482 . . . . . 6 ((𝜑𝑑 ∈ ω) → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
17 intss1 4797 . . . . . 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 9615 . . . . 5 ((𝜑𝑑 ∈ ω) → ran (𝑌‘suc 𝑑) ⊊ ran (𝑌𝑑))
2518, 24sspsstrd 4006 . . . 4 ((𝜑𝑑 ∈ ω) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑))
26 dfpss2 3983 . . . . 5 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑) ↔ ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌𝑑) ∧ ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑)))
2726simprbi 497 . . . 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 3233 . 2 (𝜑 → ¬ ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
30 fveq2 6538 . . . . . . 7 (𝑏 = 𝑑 → (𝑌𝑏) = (𝑌𝑑))
3130rneqd 5690 . . . . . 6 (𝑏 = 𝑑 → ran (𝑌𝑏) = ran (𝑌𝑑))
3231unieqd 4755 . . . . 5 (𝑏 = 𝑑 ran (𝑌𝑏) = ran (𝑌𝑑))
3332cbvmptv 5061 . . . 4 (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = (𝑑 ∈ ω ↦ ran (𝑌𝑑))
3433elrnmpt 5710 . . 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 268 . 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 207  wa 396  wal 1520   = wceq 1522  wcel 2081  {cab 2775  wral 3105  wrex 3106  Vcvv 3437  wss 3859  wpss 3860  𝒫 cpw 4453   cuni 4745   cint 4782  cmpt 5041  ran crn 5444  cres 5445  suc csuc 6068  1-1wf1 6222  cfv 6225  (class class class)co 7016  ωcom 7436  reccrdg 7897  𝑚 cmap 8256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-om 7437  df-wrecs 7798  df-recs 7860  df-rdg 7898
This theorem is referenced by:  fin23lem39  9618
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