Theorem fin23lem38 10348

Description: Lemma for fin23 10388. 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 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
fin23lem.f	⊢ (𝜑 → ℎ:ω–1-1→V)
fin23lem.g	⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
fin23lem.h	⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
fin23lem.i	⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)

Assertion

Ref	Expression
fin23lem38	⊢ (𝜑 → ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)))

Distinct variable groups:   𝑎,𝑏,𝑔,𝑖,𝑗,𝑥,ℎ,𝐺   𝐹,𝑎   𝜑,𝑎,𝑏,𝑗   𝑌,𝑎,𝑏,𝑗

Allowed substitution hints:   𝜑(𝑥, 𝑔, ℎ, 𝑖)   𝐹(𝑥, 𝑔, ℎ, 𝑖, 𝑗, 𝑏)   𝑌(𝑥, 𝑔, ℎ, 𝑖)

Proof of Theorem fin23lem38

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 7892	. . . . . . . 8 ⊢ (𝑑 ∈ ω → suc 𝑑 ∈ ω)
2		eqid 2765	. . . . . . . . . 10 ⊢ ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘suc 𝑑)
3		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑏 = suc 𝑑 → (𝑌‘𝑏) = (𝑌‘suc 𝑑))
4	3	rneqd 5930	. . . . . . . . . . . 12 ⊢ (𝑏 = suc 𝑑 → ran (𝑌‘𝑏) = ran (𝑌‘suc 𝑑))
5	4	unieqd 4887	. . . . . . . . . . 11 ⊢ (𝑏 = suc 𝑑 → ∪ ran (𝑌‘𝑏) = ∪ ran (𝑌‘suc 𝑑))
6	5	rspceeqv 3606	. . . . . . . . . 10 ⊢ ((suc 𝑑 ∈ ω ∧ ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘suc 𝑑)) → ∃𝑏 ∈ ω ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘𝑏))
7	2, 6	mpan2 704	. . . . . . . . 9 ⊢ (suc 𝑑 ∈ ω → ∃𝑏 ∈ ω ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘𝑏))
8		fvex 6898	. . . . . . . . . . . 12 ⊢ (𝑌‘suc 𝑑) ∈ V
9	8	rnex 7913	. . . . . . . . . . 11 ⊢ ran (𝑌‘suc 𝑑) ∈ V
10	9	uniex 7749	. . . . . . . . . 10 ⊢ ∪ ran (𝑌‘suc 𝑑) ∈ V
11		eqid 2765	. . . . . . . . . . 11 ⊢ (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏))
12	11	elrnmpt 5950	. . . . . . . . . 10 ⊢ (∪ ran (𝑌‘suc 𝑑) ∈ V → (∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ↔ ∃𝑏 ∈ ω ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘𝑏)))
13	10, 12	ax-mp 5	. . . . . . . . 9 ⊢ (∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ↔ ∃𝑏 ∈ ω ∪ ran (𝑌‘suc 𝑑) = ∪ ran (𝑌‘𝑏))
14	7, 13	sylibr 237	. . . . . . . 8 ⊢ (suc 𝑑 ∈ ω → ∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)))
15	1, 14	syl 18	. . . . . . 7 ⊢ (𝑑 ∈ ω → ∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)))
16	15	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)))
17		intss1 4930	. . . . . 6 ⊢ (∪ ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) → ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊆ ∪ ran (𝑌‘suc 𝑑))
18	16, 17	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊆ ∪ ran (𝑌‘suc 𝑑))
19		fin23lem33.f	. . . . . 6 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
20		fin23lem.f	. . . . . 6 ⊢ (𝜑 → ℎ:ω–1-1→V)
21		fin23lem.g	. . . . . 6 ⊢ (𝜑 → ∪ ran ℎ ⊆ 𝐺)
22		fin23lem.h	. . . . . 6 ⊢ (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ∪ ran 𝑗 ⊆ 𝐺) → ((𝑖‘𝑗):ω–1-1→V ∧ ∪ ran (𝑖‘𝑗) ⊊ ∪ ran 𝑗)))
23		fin23lem.i	. . . . . 6 ⊢ 𝑌 = (rec(𝑖, ℎ) ↾ ω)
24	19, 20, 21, 22, 23	fin23lem35 10346	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∪ ran (𝑌‘suc 𝑑) ⊊ ∪ ran (𝑌‘𝑑))
25	18, 24	sspsstrd 4067	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑑))
26		dfpss2 4043	. . . . 5 ⊢ (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑑) ↔ (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊆ ∪ ran (𝑌‘𝑑) ∧ ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑)))
27	26	simprbi 503	. . . 4 ⊢ (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ⊊ ∪ ran (𝑌‘𝑑) → ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑))
28	25, 27	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ ω) → ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑))
29	28	nrexdv 3162	. 2 ⊢ (𝜑 → ¬ ∃𝑑 ∈ ω ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑))
30		fveq2 6885	. . . . . . 7 ⊢ (𝑏 = 𝑑 → (𝑌‘𝑏) = (𝑌‘𝑑))
31	30	rneqd 5930	. . . . . 6 ⊢ (𝑏 = 𝑑 → ran (𝑌‘𝑏) = ran (𝑌‘𝑑))
32	31	unieqd 4887	. . . . 5 ⊢ (𝑏 = 𝑑 → ∪ ran (𝑌‘𝑏) = ∪ ran (𝑌‘𝑑))
33	32	cbvmptv 5217	. . . 4 ⊢ (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = (𝑑 ∈ ω ↦ ∪ ran (𝑌‘𝑑))
34	33	elrnmpt 5950	. . 3 ⊢ (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) → (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ↔ ∃𝑑 ∈ ω ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑)))
35	34	ibi 270	. 2 ⊢ (∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) → ∃𝑑 ∈ ω ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) = ∪ ran (𝑌‘𝑑))
36	29, 35	nsyl 141	1 ⊢ (𝜑 → ¬ ∩ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)) ∈ ran (𝑏 ∈ ω ↦ ∪ ran (𝑌‘𝑏)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906   ⊊ wpss 3907  𝒫 cpw 4564  ∪ cuni 4874  ∩ cint 4914   ↦ cmpt 5194  ran crn 5664   ↾ cres 5665  suc csuc 6366  –1-1→wf1 6537  ‘cfv 6540  (class class class)co 7419  ωcom 7868  reccrdg 8402   ↑_m cmap 8830

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403

This theorem is used by:  fin23lem39  10349